My understanding is that the original .303 loads used a cupro nickel jacket and that would probably scrape fouling out with each shot. It is unlikely any military would issue weapons that fouled so badly they would be inaccurate or worse difficult/impossible to chamber rounds in after just a few shots so they must had had a system that worked.
Having said that, if you use a gas checked boolit of appropriate size with a grease cookie or wad under it you may get a few rounds through the gun before fouling becomes too bad.
A pull through with a wet patch on it might work for wiping the bore every few shots... just drop through the chamber then pull the wet patch through from the muzzle.
Regardless of how well it works (or not), you will probably have fun and learn some things. I know I'd try it.
Longbow