Not looking for the legal response, looking for for the employer would react, while just happening to be an attorney.
Using their law firm email address, an employee makes a complaint to a small business owner about an unfortunately unpleasant experience. The experience is personal, and completely unrelated to their employment at your firm. The small business owner is being reasonable. The complaint is reasonable. The request, and further DEMAND is not, and the small business owner is uncomfortable with the situation, due to lawfirm@yournamehere.law
Small business gets a customer complaint about an employee (teenager) being rude. Complaint demands termination. Owner responds appropriately, apologized, will handle it. Not terminating, as it's not appropriate. Definitely will be sure to get the child to understand why they were wrong.
The customer disagrees. And uses their law firm employer company email to say as much. Politely, but still, using work resources, and your name, and this matter is at best a "hey the kid was rude to me, please teach them better" and at worst a Google review that says someone was unpleasant , but it is normally an awesome place to go.
Scale of 1-10 is this inappropriate? (1 being very inappropriate, 10 being why are you even asking)
If the business feels intimidated by this, are they overreacting?
If you find out, as their employer. As the name on the building. How do you respond to this business in a complaint to you?
Personally, this WOULD merit termination, since it's "my" name they're using to, in the business owners eyes, coerce a specific response.