First step would be a careful examination with a borescope to see how well the re-bore went.

You'd probably want to slug the barrel at a few different points along its length to see if the ID is consistent - after all, someone pulled the barrel for a reason!

I'm not knowledgeable enough about cartridges and loads to suggest a combination, but it wouldn't take long to figure out probably.

At .310 you're very close to the nominal diameter of .303 Br, so that would at least give quite a few bullet options if correct.