It looks like the original switching regulator died overnight while charging when the case was closed and it thus got hotter than before…
I thankfully had another regulator with over-temperature and -current protection:
This one does not nearly get as hot as the original one, but also has around 10W maximum power.
Concerning the low-battery behaviour: As indicated in HJKs test of the box, when the batteries run low, the 12V output voltage is” switched off”, but not really, there is always the cell voltage still present at the output port. If the attached device has a quiescent current at 3V, deep-discharged cells might be the result of a really long power outage. The output power does not come on after the batteries are charged. To make things worse, the box only works again when either charge current over the inteded USB input cable is provided for at least a short time, or the internal reset button is presse.
So it’s not really a UPS for a layman once the batteries have run down.
But as my main goal was to be able to charge off multiple voltage sources, I am satisfied with my mod ( if the new regulator lasts, that is).
Edit: The new regulator also became a little too hot for my liking, so I installed yet another one (CC CV step down regulator) with adjustable output current:
This one now works perfectly even for long charge times without overheating.