It took a lot of reading through the 600+ page plan document and finding loopholes. For example, I asked them that if they thought it should be done for $700, who would actually do it for $700? They couldn't list one provider that would accept the payment rate they thought they should pay. And since this was a reimbursement, and they had already pre-approved the procedure (knowing the price), I think I caught them already saying they would do it, but them not following through. There was a lot of going back and forth about in-network vs out of network, but again, no one in-network would perform the procedure. In the fine print of the plan document, it said if there are no in-network providers, the plan must provide you were in-network rates and deductibles for using an out-of-network provider.
So I told them if they can't find anyone who will accept what they are willing to pay, their figure is obviously wrong. It helped that with my health insurance, everything was supposed to be covered, 100% with no copay. So the fact that they tried to charge me for the $3300 overage was very out of line for them.
They never just said "no, we're not going to cooperate" it was more just endless stalling on their part. It was never really a blunt refusal, or if they tried that, I just proved how ridiculous their arguments were. I kept a lot of documentation.
I think in the end, they just relented because they were spending more dealing with me than actually just paying me my money. But it was certainly an uphill battle the whole way, and not one I would recommend to someone who isn't well versed in legalese.
See also, the TSA precheck systems. By allowing people like frequent flying businessmen and politicians to avoid the standard TSA experience, they reduce the likelihood that such people might use their influence to advocate against the TSA, for the betterment of all passengers.
See also the US unemployment benefits situation, where many employers will contest an unemployment insurance claim by default even when the worker is actually eligible, in the hopes that some workers will not navigate the bureaucracy well enough to fight back. Because of the weakness of their case, however, they will then back down as soon as their denial is challenged.
I was reminded of this just yesterday by a similarly simple story: https://twitter.com/chrisalbon/status/1172161958759395333
It's sort of like "add to cart to see the real price" but they can't say that. And sometimes hospitals are in a different mode where they fight for it. It's not logical, it's screwed up, but anyway the first bill is usually just fake.