I don't think the battery has got to do with your aircon problem. Kung lamig ang pinaguusapan, at ok naman ang freon. There's one part in your aircon system that is clogging up. Either the dryer unit needs replacement or your evaporator/cooling coils needs change.

Batteries can be tested at any battery sales outlet depending on what brand you are currently using. Best of all it's free of charge. Yes, literally free charging too if needed.