I have already ordered the P6KE250A, they should arrive here tomorrow or Monday. Of course I still have to check the capacitors on the primary side. I have already done this on the secondary side, as I initially suspected a short circuit on the secondary side. (Physically speaking, it's exactly the same with this flyback converter).
Edit:
Everything is great, the charger works perfectly again. For a 120€ device, the effort and the 27ct for the TVS was definitely worth it.
The capacitors still have values like fresh from the factory, so I don't think it's necessary to replace them yet.
I will now do an extensive test, but I have to discharge my battery first, as it is an automatic charger and only works with the battery connected. With a full battery, it always switches off after a few seconds.