My son was worried when his Miata got totaled (other driver's fault). We had just put new tires on it and had done a lot of work to it. They ended up giving us $3500 for it, and we were happy. I just told the adjuster what I wanted and didn't back down. It was easier because the other driver was at fault. The $3500 was the cheapest part of that claim anyway, even the minor injuries, x-rays, etc. came to way more than $3500. (Just bumps, bruises, sprains, etc...). The insurance company is way ahead on this one, imagine if you had broken a bone in the accident? That one thing alone would exceed the value of the car.
I would keep in mind I bet it is a lot of hassle to part out a car. I didn't even consider doing it for a second with the Miata. There are a million of them. Our cars on the other hand, I would consider parting mine if it ever got totaled. Our parts seem to go for a lot more
If you have room then I would say go for it. DOn't offer too much. Maybe when they are approaching a number you like for the value, say you will take that plus the salvage. Maybe they will be glad to be done with it give it to you. I wouldn't feel like you were cheating anybody if you get a good deal on it.
PS I want dibs on your back underbody tray and the small side one, if they aren't all scraped up.