Jaw surgery, orthodontics, Shriner's...

I would get other opinions before doing something like this.
My DD has a small mouth, they had to put in an "expander" that allowed all the teeth to fit.
This was about 20 years ago.
I have no idea if yours is a similar situation or more complex. 

Call your dental plan and ask for local references. Also ask on any city/community groups.

Dd has a high, narrow palate with severe crowding.

She had several teeth removed. Then got a palate expander. It was an auto one we didn't have to crank.

Then she got these spacers put in. Then braces with specific set up.

I know someone who did the jaw surgery and regrets it. It made things worse.

Because of my injuries from the DUI, I was offered the surgery but said no after the pros and cons list. The time with mouth wires again and the possibility that it might not work made me say no. I had my mouth wired shut for 4 weeks after my first post accident surgery and that sucked.

Shriners does not bill patients. They will bill insurance if you have it, but that's it. DH has clubfoot and has been a Shriners patient since birth (he's now 16). At no point would we have qualified for financial need, have never been asked for any financial information from Shriners, and we have never been billed a penny by Shriners.

We do make a sizable (for us) annual donation in our son's name. My parents and my in-laws are also annual donors in his name.

We're very grateful for their expertise - the care he's received over the years has been incredible.

ETA: It's possible that something has changed I'm unaware of - DS hasn't had a surgery since preschool - but billing patients would be a huge change and surprising to me. He had an extensive imaging and movement study check-up last year and they billed insurance around $8k; we were billed nothing. I don't know if there are any variations from location to location or if orthodontic treatment is any different (tbh, I didn't even know they did that).

Good luck!

Did the current orthodontist do 3D jaw/mouth scans/x-rays? It is where you sit and the x-rays machine moves 360° around you.

Do they want to do the jaw surgery to expand her lower jaw out?

Oral surgery is not covered under dental insurance. That is definitely a medical insurance claim. I don't think you could find any dental insurance company that would cover that. If wisdom teeth are covered under medical, jaw surgery would definitely be covered under medical.

Based on the info given here, I would pay out of pocket for a second opinion. It may save you thousands of dollars.

Since I was a kid all of oral surgery has been covered under insurance.

Under my dad's plan. And now under dh's plan.

That covers oral maxiofacial surgery of any type. Wisdom teeth. Extractions. Etc.

Please get a second opinion! I have ran into so many issues with dental work. Some want to do so much and so drastic - I would really be concerned about the surgery aspect of it. I have found that the larger dental practice or "big box" type practices are THE WORST! IMO, they are padding their wallets with unnecessary procedures. You will know best what to do for your own child, but get that second opinion even if it costs you something. It could save SO much down the line.

Same for my daughter. No teeth pulled, just the expander. She really had a fun gap there for a bit. LOL But the expander worked great and she was about the same age as your daughter. This was less than 10 years ago.
DH had braces, so he went in with DD on consultation, and my DH is NOT afraid to ask questions and if something is really necessary and why it's necessary and if there are other options that work as well, but less invasive. But the best part was (and this was a little pre-planned) my DD did the 'wheel and deal' on the price. She asked if she (we) pre-paid the cost of the braces, etc. - it there was a price break. And she got one! LOL We don't have dental insurance - and I am pretty sure they don't bill as much as they do if they can get money out of an insurance company.
We are far from destitute - to ask for discounts on dental. It was never an issue to pay it outright. But - it never hurts to ask for a price break!

If the jaw surgery is going to be a necessity - it should be covered under your health insurance. All my kids wisdom teeth removals, which were oral surgery, were covered on our BC/BS. Well...sorta. Our deductible is so high, we paid pretty paid it.