When you set up the CD, GM knows about it. The destination dealer's code is entered, not the selling dealer's code.
$200-$250 is customary in the business for a CD. You pay the CD because that's how it works. It doesn't matter whether it is a truck or Camaro, Dodge or Ford. Anything more than that price is starting to gouge (because it is a 'vette, right?).
BUT, this is a lot like women and hair cuts. They think they should pay the same price for a hair cut as a guy does, and some actually file lawsuits about this.

But you can bet that guys are 1000X less picky than women.
The same applies to Corvette buyers. They have higher expectations, and there's a chance they will take a bunch more of the CD dealer's time.
That's not to say there aren't people buying Mini-vans who aren't a major pain to deal with because they think the fake wood on the outside of the van should look real
I don't quite follow how the PDI could be charged to the warranty.
You say 'no-frills' PDI, but I don't believe you can negotiate such a thing. What if there are issues with the car? Sorting them out could take a lot of everyone's time: CD dealer, selling dealer, GM, and you.
What if I don't accept the car, for whatever reason? I'm 2700 miles from my selling dealer. The original selling dealership owns it - 2700 miles from their sales and service department. When I went from delivery at the selling dealership to cross-country CD, my deposit went from $500 to $3000. I don't think that is unreasonable.
I think you pay the $200-$250 and be happy if everything goes well. If they want more, consider finding a different CD dealer.