The user is able to send emails manually.
No other add-on users have informed me of the same issue.
It seems to be specific to their account.
My add-on has an option to remove authorization, which results in them being prompted to reauthorize.
Even though it seems likely that it's a setting in their GSuite account, we can't seem to determine what it is.
The account that the add-on is installed in, is a user in the domain.
I did find some information about how to restrict messages.
I'm starting to wonder if this is a Gmail issue.