If you want to go old school, then get a 250-grain SWC moving about 900 fps. If not, then you'd probably be fine with any of the heavier .45 ACP loads on Doc's list if they shoot well.
I had a 4" Model 625 Model of 1989, and I found that a roll crimp gave the best accuracy and velocity. Unfortunately, getting a roll crimp was a pain in the ass. A standard 45 ACP die will roll crimp cases if you screw it down far enough, but only after applying a very strong taper crimp. That crushed the driving band of the SWC bullets I was using and ruined accuracy.
A 45 Colt die will roll crimp cases without the taper crimp, but it's too long for a 45 AR case mouth to reach the crimping shoulder, so I had a buddy with a machine shop cut the bottom 3/4" off of an extra 45 Colt die. Presto, instant proper roll crimp. The ammo I loaded with that die shot well and did about 850 fps in a 4" barrel, which should be enough for deer. I also had to trim RP 45 AR brass to a consistent length, as it was pretty sloppy as it came. Starline is probably more consistent, but I've been out of the 45 AR game for about a decade now so I can't say.
The whole point of getting a 625 was to simplify logistics and shoot 45 ACP ammo, but it shot 45 AR ammo so much better that it wasn't worth the trouble. I ended up selling it and hunting deer with a 5" M-629 Classic, which is pretty much ideal for the task.
Okie John