Finally, I seem to have isolated the problem:
the mini temp/humidity sensor (SKU 101020074 ) on the I2C bus seems to be responsible for the trouble.
As soon as it is disconnected, the Base Hat works for hours, otherwise the bus jams after a few minutes (no matter if the sensor is spoken to or not).
As far as I can see the hat and the sensor should be able to co-exist- the address ranges do not overlap.
Could someone help solving this puzzle?