The .357 is certainly capable of killing a deer (and so is the .22lr). But it is not a optimal choice.
As you know, 1000 foot pounds of energy is often quoted as a minimum for reliable stopping power on deer. One can argue with that, but as a rule of thumb it's not bad; and in some jurisictions it is a legal restriction. FWIW, the .357's energy typically falls below that threshold.
If you can shoot your carbine accurately and limit yourself to 100 yards (greyling22 says 150; guess I am more conservative), then yes the .357 chambering would be adequate. But your chances of wounding will be significantly greater than with a more capable round.