I use syncthing too. I ended up downloading the binary linux-x64 release and making a systemd user service for it. Maybe there's a better way, but what I found on flathub looked a bit sketchy.
I'd usually just keep one synced 'saves' dir and symlink the games save data into it.
Ive never used it but maybe this will work out for you? https://github.com/mtkennerly/ludusavi/, says its built in with Rclone so you can sync to whatever cloud you want
there's a decky plugin to use google drive or one drive (it should work with any other cloud service but requires extra setting up), haven't tried it but seems to do just that