If anyone is interesting in whether the battery can be replaced, the answer's yes-ish.
As you can see in the picture, it's powered by a CR2032. The only hurdle here is that it's welded to the contacts. Not insurmountable though. Also it was glued (or ultrasonically welded) shut, I had to put the orange part in a vise (with some cardboard to protect the plastic) and carefully yet forcefully pull it from side to side. I was surprised at how much force I had to use, so be warned.
EDIT: the fix and putting it back together
Because the top battery contact was spot-welded on, I had to pry the end a bit upward with a knife so I could grab it with pliers and sorta roll/pull the contact off, as you can see below.
The battery's also glued to the circuit board, just gently pull it off making sure you don't pull the bottom contact off as well:
I don't have a picture of the next part, but it's pretty straightforward. I straightened the top battery contact so that the new battery would fit in between. Then I carefully put the circuit board back in the case, and pushed the new battery in between the two contacts (mind the polarity). It should be a very tight fit. At least in my case it was, and it was impossible for the battery to move.
After that I put some really good glue (something like Pattex Extreme Repair, not a fan of superglue for this application) on the orange top of the case and closed the case.
I used a clamp to keep it that way for the next 24 hours: