I would use the latex medical gloves as they are thin and it's easier to grip with them. Just wash them after you drop the neg into the water stop and developer won't contaminate the fix.
I use these gloves for processing larger size FB prints and just keep washing them. Or use one hand for developer and one for the fix.
I've often wondered if you could use the old "Hardener" in the fixer with lith film. That might protect it in the wash and later.

david