I used to hunt whitetails quite a bit with a handgun in Texas and Nebraska. I tried both 158-grain and 180-grain bullets and much preferred the performance of the 180s. But that was 25-30 years ago and bullets in factory cartridges are much improved. I would recommend that you use whichever bullet is most accurate at the ranges you will be shooting deer.
FWIW, I have killed a few lions and administered a coup de gras to several mule deer using 125-grain CorBons in a 4" Model 66 Smith and Wesson--never had to shoot one twice.
