The good news is that rose propagation has remained pretty much the same since well before the Regency Period and well after! I've included some text from original gardening volumes from 1808 and somewhat later, just to give you a bit of a Regency flavor.
Do not let the term Moss Provence rose throw you. The Provence rose class was the term used in the Regency to describe what was later (mid-century) called the Centifolia class of roses. Moss Roses initially developed as natural sports (mutations), and the very first were sport of the Centifolia class (or Provence). The Moss Provence may be considered to be what we call the 'Common Moss' or 'Old Pink Moss' which looked exactly like the Rosa centifolia, except with the green, mossy growth over the sepals...
The basic methods were:
Layering is basically a method where you take young (one year) canes (the best are the ones which bore flowers the previous year) and bend them down to the ground.
You dig a small, shallow trench 3" dip, scrap off a bit of the bark to aid rooting and cover with soil. Put a rock or something to pin it down so it doesn't spring back up out of the soil.
That's it. Just wait. Once rooting has occurred, the protruding tip will start to grow and the parent cane can be cut away. This generally occurs in about two months or so.
These are the natural offspring of roses. The Gallica roses, in particular, throw up suckers. These are basically roots that have traveled a few inches or even a foot away from the parent bush, and then send up new canes. When new plants are desired, you can shovel prune these by plunging a shovel into the ground between the infant plant and the parent plant to sever the connecting root, and then dig up/transplant the new plant. You can also treat them more tenderly by digging up the ground carefully between the parent and infant, severing the root with a sharp knife and then preceding to dig up/transplant the offspring.
This is also how most modern roses are propagated. It is to basically cut out a piece of the rose you want, and stick it on a cane of a common rose you already grew, just to get the developed root system (today, we use the root stock from the 'Dr. Huey' rose quite a bit in the U.S. and I believe the English often use 'Laxa').
Inoculation is the Regency term for budding. Special knives are acquired for this purpose. They are described (circa 1808) as follows: "be provided with a proper budding-knife, or sharp pen-knife, with a flat ivory haft. The haft [handle] should be somewhat taper, and quite thin at the end; which knife and hast is to be used as hereafter directed;..."
Because I think the original description of this is interesting and has merit, I'll include that here and make more comments at the end.
"Inoculate roses. This is to be understood principally of some of the curious kinds, such as the moss provence, and others that seldom produce suckers, or at least but very sparingly; for it is by suckers from the root that most of the common kinds of roses are propagated.
Therefore such kinds of roses as send up no suckers may be propagated by inoculation, and this is the proper time [July].
The budding is to be performed upon stocks raised from rose suckers that have been transplanted, from any of the common kinds [generally the Dog Rose]." [The meaning of this is that you should, the previous year, grow a lot of some cheap, common rose that can be easily and quickly produced, and which you will use as the root stock only. This paragraph is talking about getting the root stock ready to be used.]
The operation itself should be performed during cloudy weather or in the early morning or evening hours. You will have two roses: the one you wish to reproduce (the parent) and the root stock (e.g. a Dog Rose with a good root system, which will be the host).
First, you will work with the host or root stock. The host rose's selected cane should be firm and ripe, and the buds (or eyes) dormant. Trim the leaves off, but do not remove the leaf-stalks.
Stand with the cane in your left hand and the budding-knife in your right. First, draw the point of the knife along the cane, making a cut almost an inch in length, just deep enough to pass through the bark; this cut has been made horizontally in the direction in which the cane grows.
A cross-cut at the upper or top-end is made, in order that the bud may be easily slipped beneath the bark. Now pass the handle of the knife along the horizontal line previously marked out with the point, slightly twisting the handle, so as to raise the bark on either side.
Now, go to the parent rose you wish to reproduce. Grasp a likely cane with a good bud eye and hold the cane in your left hand. Cut about a quarter of an inch behind the bud, let the knife dip slightly as it passes under the bud, raising again immediately afterwards and gradually passing outwards. The bud, when cut from the shoot, should be from one-half to three-quarters of an inch long. You may wish to then cut the cane off to prevent disease from entering at this wound.
If the bud has a thick sliver of wood still behind it, carefully insert the knife between the wood and the bard at the lower end of the bud, hold the bud firmly between the finger and thumb of the left hand, then by sudden movement of the knife, the wood may be jerked out without injuring the bud. (This requires a bit of practice.)
Take the bud to the root stock and place it against the cane where you made the horizontal incisions, and bound up with cotton or bast. Leave for three weeks and then remove the cotton or bast, as the bud eye will be firmly united with the cane.
In February, the host's cane where you placed the bud should be shortened to a point well above the new bud's site. A month later, cut the cane to a point within an inch of the new bud. About midsummer, the host cane may be cut off close to the bud.
The point of cutting the host cane is to remove the host's "natural" growth in favor of letting the transplanted bud develop. You must continue to be vigilant in cutting back the host canes, most particularly the one hosting the transplanted bud. You can't remove all the canes at this point, until the bud develops as new canes on this host root system because if you remove all the host's canes, you may kill the plant entirely.
Again, you will need a "host" or root stock, such as the ever-popular Dog Rose. There is whip-grafting, wedge-grafting and crown-grafting. The most common practiced in the Regency was whip-grafting.
You basically cut off a fresh cane (preferably one that bloomed) or "whip" from your desired rose and you cut off the entire top of the root stock.
Key: It is important that the desired end-product rose stock (often called the scion stock in the Regency and now, too) be of the same thickness as the root stock, so that the inner bark of the scion can be laid EXACTLY upon or in contact with the inner bark of the root stock rose. The two, when placed in contact, should be bound firmly together as in budding.
Root stock should be planted well in advance (by many months) before required for use, so that they are established in their pots before you use them. The root stock should be kept in a close house or frame (we call this a cold frame) with heat for a fortnight before grafting, so that they are a little bit more advanced than the scion rose and yield more sap to the scion after grafting.
When the bud eyes on the scion have grown an inch or so, the ligature used to bind it to the root stock may be loosened, and after some time, wholly removed. As they grown, they may GRADUALLY be exposed to open air.
Wedge-grafting: Take a cane (1-year old) from your rose you want to propagate, with three or four buds. The top of the cane should be cut at a slant, immediately above a bud. The bottom is cut into a half-wedge shape, and a cut is made into the bottom, upwards into the exposed-tissue side of the cut. (This makes the bottom of the cane forked, with one side of the fork also stripped of bark.) A matching 'V' cut is made into the root stock, also with the outer bark removed from one side of the 'V'. The cane is then inserted so that it's exposed side is against the inside of the 'V' in the root stock, and the side of the root stock that was stripped is inside the 'V' of the cane. This is then bound with cotton or twine and sealed, if possible, with wax. The exposed top of the scion (grafted cane) is also dabbed with wax to prevent drying out.
The place where the two plants are in contact should fuse within a few weeks, and the wrapping can be removed (generally by early summer).
You may take cuttings whenever the rose cane has finished blooming. Cut canes the size and thickness of a pencil (preferably canes that bore blossoms). You may place six or eight into a pot with sandy peat or loam enriched with decayed manure. Set the pot in a greenhouse. They will root and be ready to transplant to separate pots in about 3 weeks. Keep them warm until the roots touch the sides of the pots, then gradually harden off and transplant into larger pots.
NOTE: You will want to make sure that there are at least 4 buds on the cane, and that you bury the bottom-most bud for this is where roots will form. Trim off all but the topmost two sets of leaves (generally leaving about 4-6 leaves).
My 1808 book is even more specific: In October, "observe, it is the last summer's shoots that are to be used for cuttings: let strong shoots be chosen, and shorten them to about nine, ten or twelve inches in length; then plant them in rows a foot asunder, and set the cuttings about six inches distant in the row. Let every cutting be put half way into the ground."
To pollinate: pick the male donor rose and the female recipient rose varieties. At the male donor rose, cut off one side of the bud with a knife and remove the stamens or male organs before they shed their pollen. Then, on the female recipient, remove the inner petals to expose the pistils. Remove the male organs (stamens) CAREFULLY so as not to damage the pistils. Then lightly brush the male stamens over the pistils to pollinate them.
Place a canvas bag over the flower, fastening it at the base to prevent bees or insects from messing with it. The female flower must be quite dry when you do this. (Also, I only use the term female flower to distinguish which flower we are talking about--all flowers have both male and female parts.)
When the petals decay, remove the canvas bag. Berry-like hips (or heps) will gradually swell and ripen, if this was successful. If not, it will simply rot. When the pod is ripe, you may gather the pod, slice it open and rub out the seeds. They are actually quite hairy and can cause skin irritation.
Sow them at once. In the spring many will germinate, although some will lie in the ground until the second or third year. Loamy soil (sand and well-rotted manure) is the best. It will take several years to weed through the plants and find any of quality.
The first year, they should all be de-budded (buds removed) to prevent blooming and weakening of the plant. The second year, you may begin to determine if you have any roses of any merit. Out of thousands of seedlings you may have only a dozen or so plants of any value.
NOTE: In order to have more petals, the plant has to turn stamens into petals. This means that it can be quite challenging to use Centafolia (Provence) roses as the male pollen donor, since they are simply PACKED with petals! Centafolia means 100 petals...