Validate custom domain for firebase hosting

Hi, I need to verify my custom domain with Firebase Hosting.
Why do I need to do that? - I´m using Firebase Auth as a provider for authentication and Firebase Functions to verify user and set the session cookie. In order to set a first-party-cookie I have to rewrite the function url to my custom domain (with Firebase Hosting).

The problem is, that Firebase Hosting does not accept other A / CNAME - records in the DNS settings of my domain provider. As Webflow should be always the first A-Record, how can I achieve my plan? Has anybody faced that issue before?

Best, Nils