I would advise against moving profile data.

You should set your backup program to backup the files where they reside using the advanced settings described at https://www.seagate.com/manuals/software/toolkit/backing-up/#define-backup

Not sure this suggestion will work. The Seagate software referenced in your link specifically says: The Toolkit backup will not back up Hidden files, or System files. The following provides a detailed list of file paths and extensions excluded from backup. Exclude paths:

C:\Users\UserName\AppData

Which is where Thunderbird saves my files.

What will happen if I just choose a file location on the C Drive that is not a hidden file?

You can keep the profile in the default location, but change the location for the mail folders:

http://kb.mozillazine.org/Moving_your_mail_storage_location_-_Thunderbird

Backing up just the mail is probably the most important part; backing up the profile, apart from the mail, can be done manually, less frequently.

There is also an add-on to create additional local folders:

https://addons.thunderbird.net/en-us/thunderbird/addon/localfolder/