How to simultaneously screen mirror and either screen record or broadcast on ipad (no computer)

During COVID, I have students who are staying at home, and students who are coming in. I need to be able to broadcast my screen-mirrored lesson via Zoom/ Meet to those at home while still being able to screen mirror. My venue does not have a computer and cannot accommodate one, as the Apple TV is connected directly to the projector up on the ceiling and there is no HDMI cable for a computer. I have an iPad Pro.

This seems such an obvious thing to be able to do with Apple, given its strong education focus, yet at the moment you can only do one of those - screen record the lesson beforehand and then just play it in the lesson (what's the point of students coming in then?), or give the lesson and then screen record it again afterwards (duplication of work).

If I had access to a computer to act as an intermediary, it would be fine - Zoom/ Meet from the computer, which is set up to project, and use the iPad to share content to the call. Unfortunately, that's not an option. Anyone have another workaround? (And c'mon Apple - please get your stuff together on this one!)