Can I insist on not going through insurance for out-of-network service?

I have a family member seeing a specialist out of state. We knew it was out of network but believe they can give her the best care. We told the office up-front we’d be self-paying and knew insurance would not cover it, so we put down the $5k self-pay deposit. The office wanted her business office to be sure insurance wouldn’t cover it, so they took her insurance info.

Now the estimates have gone up, and they called saying the insurance doesn’t cover anything (no surprise) and her OOP max is $20k, of which she has met $0 so far. We said that’s fine we were planning to self-pay. Then we asked why can we have the self-pay discount put back? They insist they plan to bill us at the full rate, send it to insurance, wait for the EOB, and then bill us. We told them that doesn’t help because the insurance will not cover $1.

Can we insist on going self-pay and them not running it through insurance? The two people we just had on the phone said no, but I want to be 100% certain before she racks up a much bigger bill than she has to.

Tldr; Going to out of network specialist as self-payee, they want to go through insurance anyway instead of giving us a cash rate.