@Dylan OMG OMG OMG I figured it out!
I went into TrueNAS and found the Applications dataset, found the permissions and everything looked fine there, but I also noticed that under the Applications folder there was the Plex folder. Only the Plex folder. Because I had only installed Plex, nothing else. And since I uninstalled it, it shouldn't still be there. The fact it is still there, I reasoned, could be causing issues when it tries to install. I figured it must not have fully deleted the folder during the uninstall. So I deleted that folder and then tried to install Plex again through HexOS and IT WORKED!
So happy! Thank you for pointing me in the right place!