Here's a trick I have used a few times in the past. I haven't had to do it for a few years now (since we went trade only), so I can't guarantee that trading standards still work the same way.
If we got someone being unreasonable and threatening to report us, I would call trading standards myself, and explain that I was a trader involved in a dispute with a member of the public, and looking to them for guidance. This doesn't happen to them very often, and automatically puts you in their good books.
All they will do is quote the law (which I usually knew anyway, but its always useful to have it confirmed) but I would then ask them for a name and a reference number, in case I needed to come back to them for further advice. Thank them nicely, and hang up.
Then phone back the customer and say, ever-so-politely:
"Since it is apparent that we are not going to agree on this, I have taken the precaution of speaking to trading standards. They have opened case-file reference number 12345, and the matter is being dealt with by Mr Smith. His number is 01234 567890, extension 123. Perhaps you would like to give him a call?"
None of those matters ever went any further.