What part of the cage is starting to rust? Is it the screen, the frame or both.
Most if not all screen cages to my knowledge are made out of aluminium which doesn't rust it corrodes. Rust = brown iron oxide, Corrosion on aluminium = white chalky build up. What usually happens is the water gets under the paint and then like a cancer it spreads. Once it spreads far enough it usually starts to release and fall of as chips of paint. If you look closely at the effected areas the paint will not be smooth. As far as a health risk to you cham, I doubt that he would eat the paint chips. I also doubt that you would leave them in the cage for any length of time with out cleaning them out. Now as far as the fix goes, there's only one way that I can think of. You have to sand the effected areas and repaint them. SORRY! I recently bought a used cage that was in bad shape so I disassembled and primed and repaint the whole cage. You can just address the effected areas of course matching color is always a pain. Satin paint is probably going to be the best sheen match.