I know this post is a little old, but I've been having the same issue with my 2009 Aveo for about 2 years now, and I bought it brand new, base model as well.
Chevy wouldn't acknowledge any problem once I replaced the first dead battery with a non-chevy brand battery, and I've not had the money nor energy to put into figuring out what's the issue. So I go and get my still under warranty, only a couple weeks (or sometimes days) old battery replaced all the time. I've probably replaced it 18 times in the last two years.
I'm not as auto aware as some might be, but the issue sounds the same.. battery dies, its not the alternator, and nothing appears to be putting a drain on the battery, although something clearly is.
Sometimes the replacement battery will last a few months before dying over night, other times it will only go about a week before I have to replace it again. It will always jump, but drains progressively more quickly the longer I wait to replace it. Sometimes its less than 4 hours before its drained again, other times it takes over 12. I've seen a ton of forums on the issue for the 09 aveo, but never have seen one where someone has figured it out.
kind of just waiting for the factory recall at this point, but would love to hear if you've gotten any further in this.