Same situation here, I too am running an Android phone without Google Play Services installed, mainly due to privacy concerns.
Aside from the app crashing when opening the QR code scanner, all functionality seems to work fine.
This QR code issue I can mostly mitigate by sharing payment links to the browser on my phone, after which the Bunq app picks up on it.
However, it would certainly be nice if there were some option to suppress the warnings that pop up continuously.