I mentioned that I struggled a bit on KYC level 2. Apparently, if KYC is rejected, the button 'upgrade KYC' becomes available again in the app. No further message or warning. Via a Google search I learned that the reason for rejection of level 2 KYC can be found under Account Settings. Apparently, my 'proof of address' was rejected.
I sent a copy of my tax application for vehicle tax: cannot become more official than that but apparently that is not good enough for TenX

Trouble is that all utility services are registered on the name of my partner. So how to proceed?

I used my driver's licence as here it has your address on.
As for your application- if you filled in the application I imagine it wouldn't be. Perhaps the reply to your tax application instead?