I think I first need a better understanding of where specifically in mp3tag you can edit the info that shows up as the "TV Show" name in iTunes. I've read through posts and FAQs on here that show what that tag is called, and what relevant iTunes tags are supported, but I can't find where I can change them when editing a file in mp3tag
You could load a file that shows up as you want it.
Then open the extended tags dialogue (Alt-T) and see the names of the fields.
Please note that iTunes very much likes to keep data in its own database and may not save it as metadata in the files.
Once you found the field names, take an untagged file,
press Alt-T again,
click the "new" button in the extended tags dialogue
select the correct fieldname (perhaps you have to type it in)
Enter the data for that field
Repeat that until all fields have been created
Finally click OK to save the modifications and close the extended tags dialogue.
Check what iTunes makes of it.
If this works, it may be worthwhile to adapt the tag panel and the columns in the file list so that they represent the fields that you need most frequently.