Anybody have experience with Card Benefit Services and a rental company deadlocking over diminution/diminished value?

We banged up a Hertz car over the summer in Florida. I promptly submitted everything to eclaimsline and the idiots at Card Benefit Services initially sent me an e-mail saying they'd approved my claim--for about half of Hertz's demand. When I called to ask what was going on, they said it was a mistake, and that they would negotiate directly with Hertz, and that I shouldn't owe anything. A few hours later I got another e-mail saying they'd sent a release to Hertz to confirm the negotiation and settle the claim.

But apparently the idiots at Card Benefit Services screwed up again, because today I heard from Hertz that while Hertz received and signed a release for the full amount of the demand, Card Benefits Services decided that the release was also a mistake, and they are actually only willing to pay Hertz about 2/3 of the demand.

I called Card Benefit Services again, and they say the issue is diminished value. I thought Hertz's demand was quite reasonable in light of the damage (Hertz gets great rates on parts and labor based on the repair bill they included with their demand), but about 1/4 of the overall demand was for diminution of value, and Chase says they won't pay it. I've checked and diminished value is a valid claim in Florida (not so in every state), and Chase's terms do in fact exclude it. Card Benefit Services says it doesn't matter how reasonable the rest of the claim was, they won't pay a dime for diminished value.

Hertz initially said that the deal with Card Benefit Services was off altogether and they were demanding the full amount from me. When I pushed back they said they would sign Chase's reduced release if I'll first pay the difference between that and the full claim (minus a small "customer service" discount).

Before I get my personal insurance involved (because isn't the whole point of primary coverage from Chase supposed to be that I don't have to get my personal insurance involved?!), has anybody dealt with this before?

I also find it bizarre that if Hertz would have declared the car a total loss, Chase would have paid the whole thing (I know this to be true as a family member had a Hertz car stolen a couple years ago and Chase took care of the whole thing), but because Hertz fixed it for a very reasonable price, Chase refuses to pay the diminished value.