The installation is on the D:\ drive (main storage), I've reset and verified it just to be sure, but it looks like the roaming app data where costumes are getting saved to and loaded from
The COH install and the Paragon Chat install are completely separate. The one you're seeing in roaming app data (probably AppData\Roaming\ParagonChat) is the Paragon Chat install, along with all it's data. That's what you want to move elsewhere, like D:\Games\ParagonChat or wherever.
It's self contained so you can just move the whole thing without having to reinstall. If you have a desktop shortcut or anything you might have to update it to point the right place, or you can just make a new shortcut to Bin\ParagonChat.exe in the new location.
PC does use the assets and data files out of the COH directory, but other than that doesn't modify it in any way. That's on purpose, just on the off chance that those files ever have another use again someday.
(client says it's saving to C:\PC\Costumes) is staying on the C:\ drive (OS)
Ignore that. The COH client uses legacy APIs, and it doesn't work with path names that are more than 260 characters long or have international characters in them. Paragon Chat intercepts all of its file access and redirects it using more modern APIs, letting COH think that it's installed somewhere easy for it to handle. When you see "C:\PC\Costumes", it's actually being saved in "[wherever Paragon Chat is running from]\Costumes"