Exchange Online users unable to book meeting rooms in Exchange on-premise

We have a hybrid Exchange environment and all of our meeting room calendars are hosted in Exchange 2016 on-premise.  The users that we have migrated to Exchange Online are able to see the calendars in the rooms list but receive no email to confirm or reject the booking, and the entry does not appear in the calendar.

I suspect this is because at present our Exchange Online users are treated as external (I need to fix this as a separate issue) but as a short-term workaround I can configure the mailboxes to process external meeting requests (this is disabled by default).

I can check this first by running the Get-CalendarProcessing cmdlet:

Get-CalendarProcessing meetinroom1@ourdomain.com

I can see that the value is set to False, and can set it to True using the Set-CalendarProcessing cmdlet.

Set-CalendarProcessing meetingroom1@ourdomain.com -ProcessExternalMeetingMessages $True