Allright, finally got it all solved. Replaced the strainer.
The original strainer was a one-piece unit that you just screw on, part number ending in 213/313.
It had the added downside that the clear strainer lid was facing to the wall when mounted on the pump.
There is an alternative model number ending in 215/315. It comes with a nut and can be mounted at any angle. So now the clear lid is facing the door. The replacement screwed on in 5 minutes and everything works fine now.
The old strainer for some reason did not achieve an airtight connection at the pump side. Whatever, it works now and we are happy. Not only can I now replace these things blind at 3am, I even know the part numbers by heart.
Thanks to everyone who answered!
