How would an 044 1/2 handle the .357 magnum or possible the .357 maximum ?

You are limited by the 3/4" barrel shank diameter. I personally would stay with the .32-20 case size or smaller.
.25-20 wcf, .218 bee, .22 hornet. Maybe a .32 H&R magnum.

This falls into the area of what about the next guy. The action should be fine, but age is a factor. 357 maximum is a 40,000+ psi cartridge, that’s a lot to ask with that large of a case head.

I wouldn’t hesitate to chamber it in 38 extra long, which is basically a 357 Max, with a heeled .375 bullet. Black powder pressures would be fine. There are plenty of example of larger cartridges in weaker barrels and actions.

For hoop strength a .380 case head in a .750” tenon of the 044 1/2 is comparable to .473” case head in a .875” tenon of the 44 1/2. Lots of 44 1/2 were chambered in wildcats based on the 7MM Mauser cartridge. 257 Robert’s being the most famous.

I have an 044 1/2 chambered in probably 30 Schwartz short. The differences between that and 30-30 Wesson are negligible. 357 Max straight taper to .308” bullet. I shoot with confidence at reasonable pressures. I also have 30 Schwartz long barrel for it. 30-30 case straight tapered to .308” bullet. That one sits on the shelf. I don’t have the confidence whoever made the barrel had.

357 magnum, makes it less likely to be over pressure by someone else.

You could always chamber it in 360 Rook and no one would ever bat an eye about safety.