Thanksfully it is fairly easy to present a virtual webcam that is a composition of what your real physical webcam is showing and whatever background you want.
https://obsproject.com/kb/virtual-camera-guide
https://usercomp.com/news/1413136/ffmpeg-virtual-camera-guid...
I used that back in the days in 2019 / early 2020 when it was not yet possible to share your screen on msteams under wayland in chrome and firefox.