We just went through this with a customer here in Illinois. The initial estimate would have totalled the vehicle. We helped him shop used parts and such and he turned in an estimate that allowed them to compensate him for that estimate and he retained the clear title. The accident will still show up, but the title is unbranded and never left his possession.
Then it wasn't a total.
The insurance company has no say in the title branding requirements and no say regarding retention of salvage. If the car is a total, the insurance company is not going to circumvent state law regarding title branding requirements just to satisfy a customer. They may negotiate the outcome of the claim, sure, as your example indicates. But if it's a total, they're following state law regarding the title. They can get in a LOT of trouble if they don't.