Oops,
By full metal jacket I thought you meant just a jacketed bullet and not a non-expanding bullet. My bad...
Hmmmmmm. You won't want the CCI TMJ which stands for Total Metal Jacket as the shell is thin and will still deform on impact. The Winchester 40grn FMJ is a true full metal jacket that has a thick outer shell of copper.
Thier performance on game is just a needle hole in and out, but penetration is great. Im not sure how you will be able to buy these bullets, as I am pretty sure Winchester themselves make the bullet and put it into the case for the WMR. Cantacting them non the subject would be your best bet and even if they won't sell you the bullets, they might be able to lead you into a good direction for getting a FMJ for Hornet reloading.
A 22LR wouldn't work not only due to the heeled bullet or full lead cast, but the fact that the 22LR isn't actually a .223 or .224 diameter bore. It is smaller; like .22, .219, .221, ect. Not sure exactly.
However, a 22 Mag is a true .223 diameter bore and will easily accomodate the use of a .224 with a little bullet turning, not much. A guy actually reloaded the 22 Mag with .224 Hornady V-Maxes and Nosler BT's.
Good Shootn'
Hildy