While recoil is going to be a zero factor with a rifle, I would go with 357
My experience with the 44 was that it was not as accurate as 357 or 41 magnum (which would be my choice if I was buying it and available but probably not). I am not saying 44 mag is bad, just not as much a tack driver as the other two. If its a fun gun I would think accuracy would be the prime goal.
Ammo costs will be lower with 357 and if you reload a lot less (44 take a pretty good powder charge).
I would think you can also shoot 38 special in it (per the above check as I am not into those guns so don't know much about them)
357 is a sweet cartridge for accuracy. Shade better than the 41 mag IMNSHO.
You might check twist rates and see if one is better suited than the other. Some odd twists have been kept for some rounds over the years because of tradition, and not that they were the twist right rate for the cartridge.