I would post the name here and let the court of public opinion rule on it. Send the company a link to the discussion so they can see how it is going. Do the same on any other forums you frequent. I bet that magically gets them to spring into action to make things right.
I’ve seen this exact scenario play out many times on various forums. One potentially upset customer is no big deal, a bunch of existing or potential future customers reading about poor service is a much bigger issue. The company often springs into damage control to avoid having things spiral out of control.
By the way, the better business bureau is not a worthwhile yardstick to measure anything. It’s basically like Yelp where people can leave reviews, and companies can pay to get rid of poor reviews. A lot of people think it’s somehow official or even maybe a government agency, but it isn’t.
EDIT you might also try posting your story on their social media pages, if the company has them. Same theory applies there as well.