With the Apple TV box in use, it shouldn’t matter what model TV it is connected to for mirroring, as the TV only knows the HDMI-in signal from the Apple TV box, not about what the Apple TV box does or what the iPhone does. TV1 and TV2 and TV3 should be the same for this.
iPhone has no AirPlay settings. That functionality is always loaded and present.
Apps might behave differently for in-app AirPlay controls vs. Screen Mirroring, although it is encouraged to program for auto configure to the best user experience either way. I haven’t used Zoom for this. Upcoming tvOS 17 (later this year) is said to bring big changes for Zoom and the like, where iPhone can act as camera for Zoom directly on Apple TV (announcement). Zoom for iPhone is often updated, and may have behaved differently for AirPlay in the past.
Based on these observations, the behavior that you describe makes no sense to me, and shouldn’t happen. Unless someone else chimes in with a new idea, I would suggest to contact an expert from the Apple Support staff for further assistance. Contact Apple for support and service (phone numbers) or start your support request online (chat/email/callback).