Use OBS to capture your screen and manipulate it however you want and then use its virtual camera feature as your webcam for the meeting. If you want both your face and screen you could easily do a facecam in the corner in obs

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

I know this doesn't solve your immediate problem when using GNOME, but if you are desktop curious, KDE natively offers sharing a portion of the screen or even creating a whole new virtual monitor with its own resolution, wallpaper etc that you can share.

So my typical workflow is to have it create a 1080p virtual monitor above my real monitors, and share that, since sharing a full 4k monitor makes it hard for people to read. Then I can show whatever I want on that monitor without worrying about exposing something from my main monitors.

I have funky screen configurations too. What I've done is setup a headless screen that I view over VNC on a tablet and share that. On Wayland wayvnc can do this.

OH - check if your monitor supports PBP - Picture by Picture. Some Ultra-wides have this functionality, and you can run two cables from the computer to the monitor (assuming you have 2 video outs) and the monitor itself will show up in the OS as two smaller monitors instead of one big one. If your monitor has this feature and you have two video outputs, it sounds like it would do exactly what you want.

Also, to add to what others have said, OBS will very likely do almost everything you are wanting, but if you want to be able to have your "webcam" and "screenshare" separate I have a silly solution for you... Use OBS to get the half of the monitor you want to share piped into it, but instead of using the virtual camera, pare down everything in OBS you don't need to see so it is basically just the preview and then in the meeting software share the OBS window. You can still use your normal webcam setup for your face, and a "screen share" option for your screen share, not limited to any windows (even though the thing thinks it is limited to OBS, but OBS sees the other windows you need and just takes up a little bit of border area). It's silly, and it isn't perfect, but it gets you 90% of the way there.

I always share a tab or window because then screen ratio becomes irrelevant, or am I missing something special about your setup?

Does your screen not have a built-in settings menu where you can turn off the stretching?

I'd select one window and just scale that down.