A3 Etron repair / maintenance question re battery

So I've got a 2016 A3 etron (77K miles, no major issues other than what's below) that's been throwing multiple check engine errors. My mechanic says that the purge valve and some o-rings need to be replaced, which should address one of the errors.

The bigger question relates to the second error, which is tied to the electrical system. The mechanic says the battery system is "in the red", which I take to mean nearly dead. But they also said it's possible that the car has stopped charging the battery because of the other error related to the purge valve. He thinks it's possible that the electrical system could come back once the purge valve is fixed because the system will start charging it again.

So, the question is, should I spend $1300 to fix the purge valve in the hopes that it will fix the battery? The online estimates I saw to replace the batteries were $5-15K, which is a massive nonstarter.

The A/C has also stopped working, for whatever that's worth. Any suggestions or experience would be greatly appreciated.