I wrote this earlier this year. I think it probably does what you desire if you’re using Gnome or KDE.

Thanks. It’s really intended to solve picture in picture for any application, but it looks like it does exactly was OP was requesting too.

I think I’d recommend OBS too, but I guess this is possible with an application I wrote.

If you want to install it, it is available on Flathub here.

edit: My webp video stopped loading for me, so uploaded a mirror to YouTube here