I am wondering if the alternator has been bad for the past few months and is responsible for killing both the 4.5-year old battery and the new one that lasted three weeks.
I’m not sure. You made me less than confident with Autozone’s diagnostic prowess when you said the read out “finally” said the battery was bad. So, I’m not sure if their reading of the alternator is good or not.
The 4.5 year old battery was probably on its way out anyway. A bad voltage regulator can certainly kill a battery. Whether or not your voltage regulator (that’s part of the alternator) is bad, is up in the air until you have it diagnosed by a competent diagnostician, not an Autozone salesclerk.
could the loose belt (compressor, I assume) or something else be causing the excessive ripple problem with the alternator?
No. If the VR is working properly it can compensate for a slipping belt.
From your overall post, it sounds like your car has missed a few scheduled maintenance procedures. I’d recommend that you check the owners manual and replace what should have been replaced by now. And BTW, belt dressing is a temporary solution to a very cheap problem. Just replace the belt, and check the tensioner.