jr,
While 223 to 223 AI is not impossible on a handi, it is tough. Mainly because the AI chamber on a rimless cartridge is slightly shorter than the regular chamber to the shoulder/neck junction.
If you really want a 223 AI, it would be better to start with the 22 Hornet barrel.
If you have a 223, it would be better to go to the 222 Rem Mag AI or to the 225 Winchester.
SO, for higher performance in the Handi, there is the 22-250, or one of the above rechambers.
The 222 RM AI can be loaded with 223 AI dies, with an occasional full length size with a 357 Mag die drilled through the top. With the 204 Ruger based on the 222 RM, brass should no longer be as scarce.
The Handi will be harder on the brass than a bolt gun, so gaining cases life will not be as much a positive.
Regular 223 ammunition is so inexpensive that I decided to by in bulk and forego reloading. $0.17 a round for remanufactered 55grain soft point that shoots better than I can.
Define for yourself why you think the AI is better and compare the gains to the cost of the work, which will probably be in the $80 to $120 range and the additional expense of the AI dies.
I'll provide some sources if you want to pursue further.
Neal