OK, so I have tried again. My problem in the 1st attempt was that I already had 3 authenticators enrolled, which is the limit, so they were not letting me enrol another one (but the error message was a generic error which had no mention of this).
I removed one of my authenticators and tried again. Now I got the Integrity popup, as expected. I got the error message from the app, as expected. I then tried to block Integrity API but I get the exact same error message. Therefore blocking Integrity does not work as a workaround, unfortunately.
I checked the logs but nothing useful there. Basically the same thing gets logged if Integrity check fails or it's blocked.
GOS users remain out of luck and will keep needing the dongle.