I noticed in the install.sh of GSR, that setcap cap_sys_admin+ep is called on the executable. So if you know any way of replicating something like that for flatpaks that is simpler than installing GSR manually, feel free to let me know.
You can just run the same command on the executable installed by flatpak. On my system it can be found in /var/lib/flatpak/app/{app name}/current/active/files/bin.
Checking dependencies for manual build and install of software
I am pretty new to linux so please excuse any foolish mistakes....