KBB is not bad, but there is no common idea of what “fair price” really is anyway! KBB is a consumer tool, the real value if what the car will bring is the auction price, and that is kept close.
As other have noted, no two used cars are the same, and that makes pricing tricky. There is a lot of gambling on both sides, and sometimes dealers have cars that they acquired for less than auction price, and that means they can sell them for less than usual… although they would prefer NOT to!
So buy it! If you believe that this is their real cost, why are you expecting them to take a loss? If you don’t believe them, why are you making a major purchase from liars? 15% below is beyond what I would consider “normal” error in KBB, that sounds like a pretty good deal, so why do you think you can squeeze more out of them?
This car is a unique item, there is no real “fair price”, there is only what they are willing to sell it for and what you are willing to pay for it. Yes, they would like to make a profit, just as you would like to keep as much money as possible. None of that really matters, though, unless you are comparing it to another car that would also make you happy.
Yes, you’ve made mistakes, but what really matters is what you want to do from here. I would not consider their current offer to be any more negotiable, and I would not be surprised if they sent that car to auction soon. If you want this car, then you should move soon, or find some other car at some other dealership as an alternative - I suspect that you are close to becoming the kind of customer that they try to hurry out the door because you appear serious about using up time but NOT serious about buying.