None of which matters, if Google increments a count for new mail that doesn’t exist in the new mail folder.
One option is to migrate to Google and Google apps for this stuff. Google wants your data, and they offer tools to allow that.
Otherwise, log some feedback with Apple, with enough details to allow Apple to reproduce this case and to verify what’s happening at the protocol and mail server layer.
But this would not be the first time that Google “got creative” with their implementations. Nor the first Apple bug, but you’ve indicated a similar setup doesn’t show this (mis)behavior with a different mail server.
As a work-around, disable he chime, and re-implement the chime usint some local rules that chime (just) for new mail in the new mail folder