OK. I started to just experiment with the invoke() and seems that setting attribute soapAction value to true does the trick.
I have no clue why this works. The documentation of invoke() for the attribute says
soapAction
Required? no
If the HTTP method is POST and is in fact a SOAP POST request,
this allows the SOAP Action header to be set. Ignored for GETs.
Apparently it sets some needed request header. Haven't checked which one.
Thank you for reading :)
Cannot yet mark this answer as the correct one, but will do so when it's possible.