.44 is a 100 yards or under gun for most people. To shoot reliably after that distance requires more practice.
A .223 kicks less, but requires you to be a better shot. It's a lightweight, but effective, round.
A .243 really does not kick much either. It's really a mild recoiling round even using the heavier rounds (90-100 grains) for whitetail.
I've personally hunted with a .44 rifle for many years. It's an effective round to take deer. However, my Father has used a .270 and a .243 to take deer while I used his .44 and the difference in the distance the deer travels before death is much less with a well placed shot.
Of the calibers you have mentioned, you should go with the .243 for deer hunting. It's the most versatile round of the ones you mentioned.