I've measured some of mine:
Loaded rounds with 270 gr Hornady spire points:
.396
.3955
.3955
Fired brass:
.3995
.3995
.399
My brass is Remington 444 brass necked down. My necks measure:
.0125
.0125
.013
Mine's an SSK, not a TC. How thick are your necks? I would at least size those bullets down to .376. Maybe consider turning necks if they're thicker than say .013" . I have some factory 375 JDJ brass laying around somewhere. I'll look for it after supper. I doubt it could be any thinner in the neck though.
Yes it sounds like you're tight. You may have to give up on the cast bullet idea altogether, depending on how much accuracy you're expecting. You can always size them down to .375 or .374 and they'll shoot, but probably not very well.
I don't think I'd be firing any more of those over sized bullets down that neck, unless you were sticking to starting loads. You're going to make max pressure very quickly. The 375 JDJ is right up there at about the limit of the Contender frame IMO. I would be careful about over pressure loads in this cartridge, frame stretching is a real possibility here.
I don't know that I'd ream the chamber though, plenty of chance to screw it up and have to toss the barrel. If this had happened a few years ago, I'd say to send the barrel back to TC for replacement. But I seriously doubt TC has any of these barrels laying around to replace it with. I think that ship has (sadly) sailed.