The 625 will easily shoot many, many, many commercial +P loads without any harm.
For shooting a deer, however, a 200 gr. would not be the load I'd choose; it lacks sectional density, a key factor in penetration. Assuming you don't handload (if you do, then we can talk 45 AR!), a 230 gr XTP, Gold Dot or other high-quality deep-penetrating JHP would be better than the 200 for deer.
If you have the 357, though, it's probably a better choice--not because of any intrinsic superiority, but because there are a lot more commercial loads appropriate for hunting made in 357 than in 45acp. Again, though, handloading chucks all this right out the window.