Still haven't figured what to do with stylus, but I managed to open the battery. I managed to slip a spudger between the sides and managed to break the glue.

For curious I've attached pictures of how it looks and the cells. They're 103450 li-ion cells and that has complicated things, they're rare to find nowadays, li-po cells are everywhere and you can get 6 of them for price of two li-ion.
They are mostly similar, but I'm not sure if it's a good idea to simply replace these with li-po cells 😬.

I've run into the same problem with the stylus. I've found that, poking something into the hole for the tip, at the right spot, can toggle it in and out of the "clicked" state, but I'm not sure what I'm hitting. reinserting the tip makes it click again

there seems to be a way to get the PCB assembly out of the stylus. it's in a carrier, and the carrier can move like half a centimeter out of the grip, but it won't give any further

good luck with the battery!

You can still buy new genuine Panasonic NCA103450 online on ebay if you want the battery to be original, but using replacement 103450 lipo cells should be charge compatible as well but may need re-routing since the contacts would be in a different location. In general when replacing multi-cell batteries you should ensure that they are all charged to the same voltage (or as close as possible) before installing them in the pack to avoid imbalance issues.

Obviously the cheap LiPo cells will carry higher manufacturing and QA risk compared to genuine Panasonic cells. But I've seen them used in enough multi-cell R/C battery packs that I wouldn't be overly worried about it. If you wanted to be extra safe you can also match the cells by internal resistance as close as possible but that would require purchasing more than 6 and choosing the best ones.