The rosette is now ready to be actually glued in the top; you should have a clean side, that will glue to the top in the groove:

And a somewhat rougher side that will be the top side:

If you executed all well, the fit should be almost tight but not too much, to allow a little swelling of the rosette when wet with glue and still be able to fit in the groove. Experience will tell you best, but i think somewhere from 0,1 to 0,3 mm is fine.
Also make sure that the width of it is pretty much constant; i usually get a maximum variation of 0,1mm overall. If you have more, try to sand carefully the outside and inside of the rosette in the widest sections, evening it out as much as possible. The groove itself should be absolutely constant in width if you use the circle cutter correctly.
Now you can apply the glue; it is extremely important to use hot hide glue, as it is thin enough to make the rosette swell and fill the groove perfectly, and gives the absolute best result.
First apply glue to the groove:

Then fit the rosette in the groove and apply glue on top:

By now it is possible that the hot glue is not so hot anymore and starting to gel; no panic, that’s ok. You previously put a hammer on top of a stove or something hot, and the hammer itself by now it’s hot too. Get it and press slowly the rosette with it; if needed, apply more glue on top. This operation will reheat all the glue, including the glue underneath the rosette, assuring a good joint. Plus, it makes the whole rosette swell, filling whatever gap there might be.

It’s now time to clean up the mess a little:

You can use any method you like to place pressure on it during the drying process, like a circular caul in a go-bar deck, a vacuum bag (pretty cool, actually) or a simple caul like this one:

It is just a disc of 1” particle board, melamine covered, with a ¼” bolt trough.
Place the top on it, the bolt trough the hole in the top:

Some newspaper on top:

And to close it, another disc of the same material, with some cork glued on one side, to allow for an even pressure on the rosette:

Put it on too:

And screw the wing nut tightly:

Since you reheat with the hammer to this point, make sure to be as fast as possible for best result; it does not harm if you prepared everything in advance!
Leave the thing alone overnight.
When you open it, you might notice that part of the groove is still empty.

There are different approaches to solve this: you can rout a groove not going all the way, maybe 350 degrees, and cut the rosette to fit in it, or you can rout a full 360 like i do and then fill the empty space. Using a circle cutter and a router, if find it less trouble to just do this. I use a piece of cedar a little thicker than the rosette, maybe 1.8mm or so:

You can cut it with the same circle cutter, to get a good fit without much trouble; double stick tape helps to keep the piece on the board while cutting it:

Now simply trim the ends with a chisel and it’s ready:

Apply glue, some paper on top, and the rosette caul:

If you can’t want wait another day, you can use super glue for this; if you use hot glue, you don’t need to reheat it, since the piece is so small and the operation so fast.
At this point the rosette sure does not look very handsome; better to clean it up a bit.

For this i find a plane with a toothed blade to be extremely helpful, both to clean it safely (no tearouts) and flat (it’s a plane!). I reserve the scraper for the last step.

Plane in different directions until you clean all the newspaper:

Now scrape it clean of all the teeth marks and you’re done: