Champion fan relay, odd shutoff issue.
Alternator vs battery "should not" be an issue
If you can post the destructions/ wiring diagram we can walk you through some checks
Basics are relay coil/ that is control circuit. Check the relay coil is actually getting power from the key. Then as mentioned above bypass thermo switch to ground and relay/ fan should activate and run. The relay coil/ control circuit should not draw much power, less than an amp, perhaps
Check supply power to relay contacts. This should be a major connection, like your starter relay stud, with a fuse or breaker.