Remove Primary Contact Flag or Delete Contact (without Sales Cloud license)

Summary:

Removal of duplicate Primary Contact needed

Content (required):

A Customer account Communications contact was flagged as Primary Contact but this setting is not required and is causing a problem for an integration. We understand multiple contacts can be flagged as Primary, if they have different roles, but this is not our practice. Another contact is flagged as Primary Contact.

I read doc ID 2023579.1 but we do not have Sales Cloud and have nobody in the org with the specified necessary privilege to execute the Delete a Contact step.

Suggestions on how a Communication contact can be unflagged as a Primary Contact or deleted? If this can't be done, is there a reason?