Apps locked with FaceID / TouchID are going to be problem with iPhone Mirroring.
I have some sandboxed and managed corporate controlled apps like Outlook & Teams and IT Security requires FaceID / TouchID plus a PIN Code as well as needing an MFA App to register them with the company. When task switching between mobile apps the screens are blanked out. They would block screenshots if they could. Even over FaceTime sharing the content in these apps is not visible.
Regular unmanaged apps are still trying to use FaceID / TouchID to unlock. My bank does that. I would have to close iPhone mirroring, pick up the iPhone unlock the Bank app then open iPhone Mirroring and that worked. But yes, it's tedious.
There is no solution at this time.