Not sure a finish reamer would do it anyway. SMLE’s have a reputation of oversize chambers so a commercial finish reamer might not be large enough to clean things up.
Obviously you don’t want to in any way enlarge the chamber itself.
First thing I’d try would be to simply (but very carefully) cut the burr down with a Swiss needle file. Granted getting at things and seeing what you’re doing might be difficult.
Or maybe get a long, round file like a chainsaw file with as fine a cut as possible - grind it down and/or tape it up so only a small bit of the cut is exposed. Then use that small bit of cut to cut the burr down.
Alternatively, maybe get an empty .303 case, attach the rear of it to a rod, glue a small piece of wet-and-dry paper to the side of the case in the proper location, and feed the case into the chamber and work the paper back and forth over the burr. You might actually be able to much the same using an abrasive rather than the paper, though I’d be worried about enlarging the chamber.
That’s all I can think of.