My teammate asked me to add him as a submitter to our submission, but I don't know if that would count as updating the game and therefore violate the rules. Would that be okay or not?

Submissions will not be reopened. Mentioning the credit in the comments might be best answer for now.

1. Go to your games page.
2. Click ‘Edit game’ on the top bar
3. Near the save button there is button called ‘More’ and there you find button called ‘Admins’. Click it.
4. Add your teammate as admin
5. Send them the confirmation link and have them accept it.
6. Tap the ‘Display as contributor’

They should now show as developers and be able to edit the project. It should look something like ours in the end.