Sure, sometimes you get lucky with insurance company body shops.
In many cases, insurance companies do a deal with the body shop that requires the bodyshop to suck up a certain amount of hidden damage. Let's say, for example, that the estimate is $1k, but once the panels come off, there is another $500 in damage. If the agreement is 20%, then the bodyshop would have to suck up $200 of the hidden damage and will get paid $1300. They have to make that up somewhere.
When I used an insurance company bodyshop 15 years ago, they used non-OE fasteners, duct taped the bumper absorber together and used a repair kit for the headlight, but charged for a new headlamp. Ultimately, another bodyshop fixed it and the insurance company charged the re-repair to the original bodyshop.
What you are referencing is a DRP - direct repair program (sometimes also called direct repair partner). In my former career, I was an I-CAR certified estimator for an independent shop that had DRPs with State Farm, USAA, and Progressive. State Farm and USAA were easy to deal with. Progressive was a nickel and dime relationship that got more and more "dime" as time went on. State Farm would allow me to upload images of damage and they approved EVERY SINGLE one of my estimates including supplemental.
Some insurance companies actually have back-channel ownership of independent shops. All-State used to own Sterling but sold it based on increased scrutiny, so it's not as common as it used to be.
My advice is:
1. Go through their insurance - don't bother with letting yours do the work for you. Even when rear-ended and not at fault the insurance company still dings your driving record (for their actuarial and underwriting calcs) for a period of time. One is usually not a factor; having a series of "not at faults" actually can cause insurance rates to go up as it you are deemed to be a poor defensive driver (crazy). Best advice - in all seriousness - spend $25 and buy pizza for the team working your vehicle at the shop. Everyone from the estimator (who can help you overcome BS with the insurance company), to the bodyman will run through walls for you. I promise you. They'll go out of their way to make sure
you get the best treatment. It'll be worth the $25.
Going through their insurance is easy as long as you stay on them. Don't let them take an inch!
2. Demand a rental.
3. Demand OEM replacements for any non-custom items. They likely will try to integrate used/LKQ. If they deny after listing aftermarket (meaning copies) items, remind them that their party was at fault. Squeaky wheel gets the grease and the adjuster working the case likely won't want to deal with the "headache." If you cannot avoid aftermarket because of mileage/age/value, demand CAPA certified.
4. Have your receipts ready to go - they will want them for the custom pieces. Include labor costs. You're better off getting a check from them for the custom parts if they have a 4 mos. wait.
5. If any paint work is necessary, demand adjacent panels be blended. This is costly for them to do, because you have to R&I handles, masking, etc. If your panel is not blended, you'll run the potential to see a difference in color match no matter how precise because of the fading that can happen. We had an issue with one car because he parked the same direction in the same spot and one side was more faded than the other - we had to bring in a machine to formulate the tint mixture on each side and then combine both for the middle to make it look ok.
6. Be sure you are not hurt. People get sore days after. Seen it too much dealing with collision repairs.