Hi @Alexander H,
Thank you for posting to Microsoft Community.
Based on your description, I understand that user is prompted for password when delete an appointment with room mailbox.
Please make sure the employee has Full Access permission to the resource mailbox.
And ensure the employee has the appropriate delegate permissions for the calendar.
Also, please kindly try to delete the appointment via OWA. If Outlook Web Version works fine, it most likely the client issue, you may try to create new Outlook Profile as following link to fix the issue Create an Outlook profile - Microsoft Support.
If there are anything else you need help with, please feel free to contact me.