cooling fan

There should be a thermo switch on the radiator, which tells the cooling fan to activate, in conjunction with the cooling temperature sending unit...so you need to verify that the cooling fan (and there may be 2, one for general cooling, the second for A/C activation) is coming on when your temperature, approximately, rises above halfway. You are definitely on the right track, but remember...even a fully functional fan (as you said by testing), will not necessarily turn on, unless the cooling system circuit is operational.... Well that's my best....good luck. .

As for the battery dilemma, you need only insure that you have a well charged unit (recent is always good) with very secure CONNECTIONS! The entire system will see an additional load when the A/C is activated (with its dedicated secondary fan), and similarly, the system will see a bit of a draw when your cooling fan kicks on....but with a vehicle that is properly charging, and a fresh battery, there should be no issues. Fans should come on as previously described!

Check the radiator fan relay?

It's a dual speed fan. Both the high and low speed relays are good. I just switched the temperature sending switch or whatever it's called below the thermostat, so we'll see if that does it. Thanks!