Wanted to follow back up on this -
I did end up trying one of the Chinese batteries off Amazon. Looked identical, specs identical, but the Variax wouldn’t even come on at all when it was plugged in. I was able to return it after a lot of back and forth with the seller, so at least it worked out alright, but I think as everyone has said, the non-Line6 batteries are a crapshoot. May work, may not.
Right around the time this was all going on I happened to get an email from Sweetwater that they were in stock so I grabbed one. I feel like I got very lucky.
This is a real issue for those of us that invested in these guitars. Maybe Line6 should look at building an adapter that would let you use off the shelf batteries in these. Or at a minimum get control of their supply chain - using a proprietary battery that is impossible to get is not a good look for the company.