Win the fight, lose the customer
Does it really matter if you're right?
Given the choice between acknowledging that your customer is upset or proving to her that she is wrong, which will you choose?
You can be right or you can have empathy.
You can't do both.
It's not the nature of capitalism to need to teach people a lesson, it's the nature of being a human, we just blame it on capitalism. In fact, smart marketers understand that the word 'right' in "The customer is always right" doesn't mean that they'd win in court or a debate. It means, "If you want the customer to remain a customer, you need to permit him to believe he's right."
If someone thinks they're unhappy, then you know what? They are.
Trying say this to yourself: I have no problem acknowledging that you're unhappy, upset or even angry. Next time, I'd prefer to organize our interaction so you don't end up feeling that way, and I probably could have done it this time, too. You have my attention and my empathy and I value you. Thanks for being here.
If you can't be happy with that, then sure, go ahead and fire the customer, cause they're going to leave anyway.