They replied to my ticket with enough info to give me the clue I needed.
Well, the fix was simple. Just follow the flow and RE-authorize The Bat! to access the Google account. Makes sense now. They support calendar so the authorization has to show that you're giving them access to the calendar. Once you do that, it does not pop up again.
So, you HAVE to give The Bat! access to the calendar, but you don't have to USE it.
Ideally the calendar would be disabled by default and the first time you enable it, THEN you go through the re-auth where you see that calendar is now on the things they can access list.