Tanker1983;
Your question brings to mind the reasons why many of us cast our own bullets and reload. As mentioned above, the 581 will handle any reasonable .357 load you want to put through it.
As also mentioned, it is a fixed sighted handgun and must be used with a load that will shoot to the sights FOR YOU (different people with slightly different shooting styles may see the sights differently).
I would favor a home cast bullet of about 160 grains with a "Round Flat" configuration. Lee Precision's bullet is an excellent example of what should work well for you:
Lee 6-Cavity Bullet Mold 358-158-RF 38 Special 357 Mag 38 Colt New
It has a large meplat (bullet nose) that should give excellent "smack" and driven at a decent velocity should give good penetration.
Since you apparently do not "yet" cast bullets or reload, there are commercial loads that use a similar bullet.
Here's one with a slightly heavier bullet that just "may" shoot to the sights (the only way to tell is to try them):
https://www.buffalobore.com/index.php?l=product_list&c=162
FWIW
Dale53