It does not need to be a .357 mag load but it would be fun to explore that possibility.
I do have a small supply of Speer GD 135gr +P .38 Special that will probably work just fine. It won't take many rounds to see if it shoots POA or close to it.
I also have some Hornady Custom 158gr XTP but I am pretty certain they would over penetrate. I think I will save those for my lever gun.