using bookings email address as full mailbox

I have created a bookings page. The bookings page then gets an email address.

In the config of the bookings page it is possible to assign a license to this booking page mail address so that it becomes a full mailbox. After doing this I can login via web to the mailbox. I can handle all email in the mailbox of the booking page and I have direct normal access to the calendar.

If I would use a different mailbox then the mailbox of the booking page, then I would need to add this mailbox as staff. The calendar of that mailbox will then be the leading "staff" and I can add it as "team member" like below.

In line with that I am now added the mailbox of the booking page itself as staff. But there is an unexpected result. The staff is set to guest and can not be changed. It looks like this:

This seems to give all kind of limitation - of which the most visible is that the "events on the office calendar affect availability" is greyed out. This seems not how it should be. How can I make that I can set this bookings page mailbox to "team member". Does that require another type of license or do we need to make changed is the office / exchange admin settings?