Distances/ability
I think everybody shoots a carbine better than a handgun, especially at distances past 50 yds. The exception might be hand-rifles, ie scoped handguns. There's no mention of sights or hunting conditions in the OP.
For me, handguns wear iron sights and handgun range is bow range (under 50, ideally, under 25).
If I just had to kill a deer, my choice would be with the .357 carbine. For sport, I'd set the .44 hunt up like a bow hunt.
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