Having fought the same mess issues in various places within our house I am always coming up with ways to battle it. The best way I've found to deal with their mess is newspaper.
Try putting up some kind of peg board, decorative pin board or something that you can pin newspaper too. Newspaper is cheap and easy to replace on a daily basis. We have feeder cages we use underneath our hanging perches in the living room. You could replace hanging sheets of paper in minutes. I keep grocery store baggies available to stuff it all in.
I have a table cloth on top of a coffee table, followed by one layer of newspaper, then the feeding cages. Inside those cages I cover the lower tray with one sheet, then the grate inside the cage gets a sheet. I do the same in their bird room cages. Each area takes me about 5 minutes to change out. I got tired of scrubbing dried bird shit off of everything. Once it dries it's a nightmare.
That's my idea.