If it's not strata, then no, Fair Trading can't get involved.
However, there are laws about operation air conditioning, including that it can't be heard in a home between 10pm and 8am. (See THIS) There goes your 24-hour gym. Also, they would have needed Development Approval from council to install the air con – did they get it? I”d be very surprised. So that's another phone call you need to make.
This is all stuff your Strata Manager should know. Call number three would be to ask them why they didn't give you this advice and how they can justify their fees. Sounds like they are more interested in looking after the retail owners than the people who pay their bills.