I'd second the suggestion to go with JES reboring. He did a Sedgley Springfield for me that had a bad bore. I had the rifle back in 10 days. He went from 30-06 to 375 Whelen. Jes recommends .200 over bore size for a minimum diameter at the muzzle. With that in mind a .375 would require a minimum muzzle diameter of .575

I would look at any of the wildcats on the 303; .338, .35, 9.3, .375 or even .411. All would be excellent choices. They all would be easy to make from standard 303 brass and CH-4D will make dies very reasonably, if they don't already have them in stock. They have a great variety of wildcat dies.

The 9.3x303 might be a fun wildcat to play with. Should run close to 9.3x57 ballistics and be a real fun round. The 375x303 with the old Hornady 270 grain round nose would make a real thumper.

Good luck with your project.