The ac does not run on the battery, but when running will cause additional load on the alternator which in turn also charges your battery. So you more likely have either a defective battery if over 5 years of age, or a failing alternator, in which case when under heavy load (with a.c. running) the problem becomes more evident. You can have your battery and alternator and charging system checked with the vehicle running by a good service center, or an alternator rebuilder/ automotive electrical repair specialist. There are alternator rebuilding companies who can test both and tell you what you need and generally fix you up for less than if they sell the parts to a service center who in turn mark them up again and sell to you. I removed and replaced the alternator on my 2001 RX myself, had it tested and rebuilt with warranty for under $150, but there are a few tight spots to get to on the RX300 unless you are patient. You might also start by inspecting and cleaning battery terminals and connections.