Exchange meeting rooms troubleshooting

Try to move exchange meeting room to another database. Add meeting room mailbox into outlook to see whether it receives meeting requests, to do this You need to get proper permissions (owner):
Add-MailboxFolderPermission -Identity room:\Calendar -User domain\user -AccessRights owner

If You need to book meetings in external exchange meeting rooms remember to synchronize GAL and check\set the below:
Get-Mailbox room_name | set-CalendarProcessing -ProcessExternalMeetingMessages $true