The old cotton mop 'n bucket type don't work that well, although I have used them years ago applying base Whitener on large new & repainting old backdrops. They are great for slopping type work but you end up painting yourself & flicking paint everywhere. This why your quids in with the flat sponge foam floor mops  for large flush surfaces & they don't dribble everywhere.

'Anovver fink' with your project which can only help, is get a hold of a friendly experianced theatre techie to assist in advising [free 'n grattis of course] on the logistics, while it is in the these early stages. Rather than leaving it untill it's too late & costly to have to fix up any 'Major Blues' to make the space workable later.

{As in the old adage of the 5 P's [Which I think it's actualy 6 or 7?]}