Doctor's office billed the wrong code (treatment, rather than diagnostic) and my PPO won't pay the $300. Should I pay it to protect my excellent credit score?

I've gone round and round for almost 2 years with this stupid office and I don't think I should have to pay because their billing incompetence. However, they are now threatening to report me to credit bureau. My credit score is very high, so I don't want to damage it over $300.

Can my credit score be damaged by $300 ding - and by how much?