Let's get this out of the way: IMO any carbon-fiber bike that's actually been run over by a car would probably require more testing than it's worth to make sure nothing is cracked - think "close-up x-rays of the whole thing, then ultrasound examination to be sure".
That being said...
What is the right approach to ask for what is needed?
The proper approach?
First, were you injured?
If yes, SHUT UP, SAY NO MORE TO THE INSURANCE COMPANY, AND GET A LAWYER
Let me repeat that:
SAY NO MORE TO THE INSURANCE COMPANY
GET A LAWYER
The insurance company's goal here is to minimize their payout. Period. If that screws you over for life, they won't care.
Why shut up? Because even little things like responding, "I think I'm OK" to someone asking you how you are after you've been hit will be used against you. If you have been injured, hopefully you haven't said or signed anything that hurts you going forward.
(I've been there - and it's really interesting how things play out. Innocent little things you might say that you think help you can actually wind up being terribly damaging to your case. The defendant in my case is probably still wondering what she said in her deposition that sunk her defense faster than the Titanic. It was a simple 6-word sentence that she thought would help exonerate her, but it did the exact opposite.)
If you weren't injured, and it's an expensive bike you can't afford to replace, see the above about a lawyer.
If you can afford to take somewhat of a loss on replacing the bike, go without a lawyer. The insurance company probably won't try to screw you over too hard, because that would prompt you to get a lawyer and cost them a lot more money and time. Who knows, you might get lucky and they replace your bike pretty much with no questions asked.
And never forget your state's (assuming US) insurance commission or other oversight office usually has some way of filing complaints if you're not satisfied.