No request is too quixotic; quixotic is what gun nuts are all about. Finding a "made up" barrel ready to screw into a SMLE in any caliber other than .303 is kinda tough. Getting one made is kinda expensive.

What about finding a .303 barrel with a wrecked bore and having it rebored to .375? There used to be many BSA-made .375s on SMLE and "Lee Speed" actions; a nice medium rifle. I think that the cartridge was called .375 2 1/2" BSA but I may be misremembering.....

Could you rebore the existing barrel?