In my reading over the last ten years or so regarding hunting with a .357 rifle, a common thread seems to be using a 180 grain bullet.

I know that a 180 grain bullet in a .357 revolver is considered to be big medicine for larger game that fights back.

There are heavy bullet factory loads available.

I would use the heaviest cast bullet that I could as long as I get good accuracy. I have a 200 grain gas check mould for the 35 Remington that might show promise if the twist is right.

I think it will work just fine at the range you suggested. Deer probably won’t know the difference between the .357 or 44-40. 
I have loaded lots of heavy bullet .357 for a modern 73 levergun. 11g of 5744, small rifle primer, with RCBS RNGC or Lee FNGC at 1.59” OAL yields good accuracy at about 1080fps. Both bullets are 200-205g finished. My barrel is 24” long and 18.75 twist. Never shot a deer just lots of steel animals. You likely won’t be as restricted on OAL or a weak action like the 73 giving lots of options. 
The bullet options in 158g range are huge and out of a rifle you will see 200’ish feet per second more speed than listed data for handguns. I agree, it will be nice to hear what worked next season.

Since getting this rifle I've tested a lot of different loads in it using 4227, 2400, Unique and Accurate #9. I've settled on the Lyman 318315 gas check cast in wheel weights at 215 grains. The rifle has a one in sixteen twist and stabilises these bullets quite well. Eleven and a half grains of #9 gives the best accuracy closely followed by seven and a half grains of unique. Velocities are 1300 and 1100 fps respectively. Both shoot MOA at 100 yards and are more than adequate for the task at hand.