I have a few lever action and single shots in handgun cartridges. They make fantastic deer rifles. And also inexpensive rifles to shoot if you have to rely on factory ammo.
A 357 in a rifle is completely different than a 357 in a handgun. And if you buy factory ammo it can be cheap to shoot with 38 Special loads. So the 357 has the advantage in cost of operations.
Personally I like the 44 magnum better for hunting. But it will cost more to shoot.
IMO the 357 nor the 44 magnum is limited to 100 or 125 yards. But after 100 yards the 44 magnum is a much better choice. But I have no problems extending 44 magnum to easily 200 yards and beyond. You just have to know your rifle.
I also like the 45 Colt in a lever action. You can push a 255 grain bullet pretty hard and do terrible damage to deer. Even with zero expansion you have a 1/2" hole going in.
Personally I have shot deer with my 44 magnum out to about 225 yards. You really have to know your equipment to do that. But you are suppose to know your equipment. I shoot only cast bullets that I cast in my lever actions. That is how I can shoot it so cheap. I can reload about 1,000 shells for $100. You can't beat that. And when you put a few thousand rounds down your 44 magnum lever action a 200 yard shot will not scare you at all.
Below is a picture of one of my 44 Magnum's in a Marlin 1894. That Marlin 1894 is tough to beat. Along with a Low Wall in 45 Colt. And you would not believe me if I told you how far I shoot that rifle. Tom.