That is unfortunate that you run into issue. You're the first one to report this issue, and since I've done the same migration myself on several Mac's, I can safely say this is not caused by Rename My TV Series (RMTV).

My guess would be that RMTV is having proper clearance to access certain files.
Especially after migrating from an older macOS to a more recent macOS, macOS will ask for you permission/confirmation that RMTV is allowed to access and rename certain files. Either you did not see the question, or you may have denied access by accident.

Either way .... RMTV should probably be listed under System Preferences - Security & Privacy - Privacy -  Files and Folders.

Make sure all the (needed) checkboxes are checked (you may see more than just these 2 - for example "Desktop", "Network Volumes" etc).

If RMTV does not show, or no options are listed, then you can try adding it manually to "Full Disk Access" - I've noticed Monterey not being very consistent when it comes to asking for permission, and it doesn't allow you to add items to just "Files and Folders".

Open System Preferences - Security & Privacy - Privacy -  Full Disk Access and click the padlock in the lower left corner and unlock it. Click the "+" button under the list of items, and select "RenameMyTVSeries" from the Applications directory.

Please let us know if this did or did not work - I'd be happy to help and your findings may be useful to others 😁