The biggest issue is a pm could work however would require the category to remain attached to the topic. In a PM however tagging is only available and will not work in this context.
This email triggers creation of a PM to a group in Discourse, with the topic title being the title of the email and the topic body the body of the email. The user is staged via their email address.
The group can then reply to this PM topic as they see fit, and emails will be automatically sent to the customer.
Customer can reply to the emails generated, and their email replies will arrive in the PM as new posts by that staged user.