I didn’t consider that an option because whenever I searched for setcap and flatpak, most threads were pretty dismissive and told OP that flatpak is made with security in mind so doing that isn’t supported.
Regardless, I tried it just now, but the password prompt (image below) still shows up when launching the autostart .desktop file I created. The .desktop file launches a script I wrote, which in turn actually starts GSR through flatpak, in case that changes anything.
Do I assume correctly that this prompt might be gone if I set the capabilities of /usr/bin/flatpak? It’s not something I want to do, so I’ll probably keep trying to get the manually built version installed.