Hmm - each site has unique client ID & secret, which is set appropriately.
It's possible there's an expiration setting - the Google Drive API talks about it being an hour, however, that may simply be to be using their credentials creation console or drive itself and not necessarily when using it via the API. Possibly worth looking into, though WP-cron's most often rate, hourly refresh, is right at the same level. So if "credential refresh" proved to solve it, might need two tasks running every hour which are offset by 30 minutes each to guarantee meets the hourly requirement.
One question might be - what happens when we click Backup Now - does that perchance peek at whatever our current Google credentials are and perhaps override the original oauth - that's the only thing I can think of based on your suggestions that could be creating the bug. Alternatively stated, should we only click backup now when also logged in as the Google account we used to create the client ID / secret?
We're taking a look at premium - the migrator addon is perhaps our biggest need at this time. Will advise if we need to transition this thread to your paid forums due to WordPress.org guidelines.