I actually updated my iMac Pro to MacOS Monterey and it's still there. Another bit of information which I think will help to pinpoint where the problem is.…
Here are the tests I've done on my end.
iMac Pro = MacOS Monterey
iPad OS = iPad OS 15.1
I ran a test thinking it was QuickTime – I both recorded and live streamed and heard that weird beeping noise (both in my video recording and also my live stream). At this point I was thinking maybe it's QuickTime causing the issue but now I don't believe it is. I have another recording software called "ScreenFlow" and ScreenFlow was picking up that weird beeping noise as well.
When I use my older iPad Pro from 2018 none of the above happens. This tells me that it's not MacOS, it's not iPadOS, it's not QuickTime and has to be something with the newer iPad Pro using the M1 chip. There is no other explanation that I can think of.
Perhaps there is a setting on the newer iPad Pro which is causing the issue?