Wrecked Third Gen Tacoma - Diminished Value check?

Here is the problem with that: Insurance companies started writing into their policies that they reserve the right to use non-OEM parts, especially if the vehicle is over a few years old. Most of the big companies here in Reno do that and you have to pay extra for them to use OEM parts.

 

I had OEM parts put in.

 

Technically, your claim is against the other party. They wronged you and are liable for your damages. They have a contract with an insurance company to insure all or part of their liability. What that coverage is becomes their business as long as legal mandates are met.
In other words, you can sue them for any damages (including a rental car) and get a judgement against them. What their insurance picks up would then be between them and their insurance company.
As a practical matter you would have to decide if all that was worth the time and trouble.