Problem solved. Thanks for your suggestions, Gentlemen -- they got my brain off the mark and moving in the right direction.
I had never considered exactly how .22 rimfire ammo is assembled but, upon close examination, it appears that they use a roll crimp. That's completely logical and most of you probably already knew that, but I didn't. Now I do. So what I needed wasn't a seating die, but a flaring (or de-crimping) die.
So I scrounged around and came up with a length of 1/4" steel rod. I lopped off a piece about 3 inches long, chucked it in the drill press, and used a combination of grinding and filing to put a 60-degree taper on one end, tapering down to small enough to easily fit inside the mouth of a fired .22LR case. Using that and a light (maybe 8-ounce?) ball pein hammer, it was just a matter of tapping my flaring tool in far enough to allow the bullets to start. Once there, I used a small bench vise to press them home. I did a set of six. One got flared a bit too much and the bullet is slightly loose. If it falls out, I'll Super Glue it in place.
If anyone else feels masochistic and wants to pursue the same project, the source for the bullets is North American Arms:
https://northamericanarms.com/shop/parts/cbb2/
They're sold for cap and ball firearms, but they are heeled bullets. They sell them in packs of 50, 100, or 250 bullets. They're all 30-grain bullets, which are too short to be .22 Long Rifle. They're closer to what's used in .22 Short, .22 Long, and some of the hyper-velocity Long Rifle loadings.
I suppose to replicate CB caps I could load up another set and then grind the bullets down to the CB cap profile.
https://en.wikipedia.org/wiki/.22_CB