I see that you have had a lot of responses.
Here is one more, also based on experience and observation on many animals.
We have shot quite a few smaller antelope with a 222 in Africa. The rifle had a slow twist and we used 50 and 55gn cup-n-core bullets. Be careful ! I've seen such bullets on impala sized game and while they were able to bring down the antelope they were not impressive and we soon limited our son to shooting spur-wing geese, oribi, and duiker with the 222. Yes, the 222 was impressive on 35-40lb oribi. Sending a cup and core bullet even faster than a 222 is not much of an solution because penetration will likely be even more of a problem.
If you want guaranteed penetration, then you need to get some mono-metals.
And with a .224" barrel you need guaranteed penetration. However, twist will require very light bullets at very fast velocities.
The Barnes bullets are too long for reliable stabilization, even the 45grain. Instead, I would recommend one of two speciality bullet makers. (You can try Barnes when you have rebarreled with a faster twist 22-250 or a 243 or 257 Roberts.)
GSC lists a 35 grain HV bullet in .224". They are listed as "in stock" so I would immediately order a box or two. Get a fast powder like H322, a magnum primer, and expect velocities above 4200fps.
As an equally good alternative, you can order some .224" 44gn HammerHunters. They should also stabilize reasonably well. The website recommends 12.5" twist or faster, but get a box and try them in a 14" twist. They should be fine, at least they give a 1.22 stability factor with a 14" twist. If your kids can see the deer, they should be able to drop it and maybe even have an exit wound.
Of course, if you are serious about kids using the rifle for deer, you owe it to yourself to consider a re-barreling with an 7" or 8in twist. Then you can use monometal bullets in the 60-70grain range and you will have a bonafide deer rifle, at least as far as a .224" can be bonafide.
Myself? Reload special bullets this year. Then, assuming that the kids would like to do more hunting, re-barrel the rifle to a 243 or 257Roberts depending on the magazine capability of the rifle receiver. If in doubt, consult your local gunsmith.