I don't quite understand what I did, but apparently I fixed it!
There wasn't anything of real value in the Process Monitor logs so I abandoned that. I went into the installation folders and just renamed all the .exe files to some similar name (opera_autoupdate.exe -> opera_updater.exe). I tried that and that didn't really work either.
I had read elsewhere while looking around that it seemed to happen to some people's Opera GX when they got a new update, so I just decided to uninstall and install an older version. For some reason though since this has started every time I open the installer it gives me an option to "Upgrade", which seems to never work as every time I open it after I "upgrade" it just tells me I can upgrade again.
This time it didn't though! I tried going through the usual update loop and it just...opened. I restarted to make sure it wasn't a fluke and apparently it just works now.
Thanks so much for your help. Again, I'm not 100% sure what I did but I certainly never would've thought to rename stuff so I'll assume that was what helped.