I can confirm the WiiM 12V trigger current is too low for a normal relay. I tried
this one which should have a low current requirement and it did not trigger the relay. An SSR
like this did work when tested switching a 230V ligthbulb by the 12V WiiM trigger but these are normally used to switch high current resistive loads, not amplifiers. They have some leaking-current/voltage when off (see
here and the linked video in that thread) so I don't want to connect my amp to it as the goal here is to save power..
The Xantech mentioned above looks like a nice solution but it is very expensive and I cannot find a 230V version. After some more research I found this:
https://www.electroschematics.com/low-current-relay/. This seems to be the most elegant solution, a mechanical relay (so 'off' is really 'off'), a very low required trigger current and no extra DC power supply needed. I ordered the parts and put them on a breadboard and it worked! So I made the schematic into a PCB with EasyEDA and ordered it at
https://jlcpcb.com/, 5 PCB's for only a few Euros. They will arrive in 3 weeks because I'm on a holiday first and chose the cheap but slow shipping option from China.
When I'm back I will test it and report here if anyone is interested in building it.
(The high voltage relay - I use
this one - is not pictured in the 3D model but you can see the outline on the PCB)