I'm not sure I see what the need for FlareSolverr is, as I understand it it's a reverse proxy that will add the cloudflare authentication cookies so that your *arr apps can access prowlarr, however you should just be able to stick them on a named bridge network and access them internally by container name.

I think using FlareSolverr is adding unnecessary complexity as you shouldn't need to make an internet round trip for containers to access each other.

I have all my internet facing stuff behind Authelia with 2fa but it still works because the containers talk to each other on their own network directly.