OK, I figured out a way. It's not pretty, but it works pretty well.
First, I set up hot keys in OBS. Go into settings, then hotkeys. For each scene, I set up a hotkey. So for my webcam capture scene, where I'm just capturing my web camera, I chose F1. For my second scene, where I'm just capturing my desktop, I set the hotkey as F2. Then with the third scene, where I have my video in the lower right corner of the desktop capture, I chose F3.
I installed xdotool (sudo yum -y install xdotool). I figured out that xdotool search --name "OBS 27" would find the parent window of the OBS app. Originally, I was trying to just do xdotool search --name "OBS" and I got multiple window IDs. I think it's because OBS has multiple stacked windows for each of the dockable windows like Scenes, Sources, Audio Mixer, Controls, and so on.
Anyway, I then figured out that I could run:
/usr/bin/xdotool search --name "OBS 27.2.1" key F1
to pass the F1 key to the main window of OBS, and it would use that hotkey to toggle to my video capture scene. Then:
/usr/bin/xdotool search --name "OBS 27.2.1" key F2
would toggle to scene two, my desktop capture scene. Finally, I added:
/usr/bin/xdotool search --name "OBS 27.2.1" key F3
to my streamdeck to toggle my combined desktop capture plus video. It works perfectly.
I hope this helps someone else!