Price/negotiation question

Looked at a used car today that I have some interest in. I have the option to sell my car to Carmax or have the dealership match the price that Carmax gave. This dealer also charges a pretty significant fee for “reconditioning” the vehicle and document processing fees. The combo of those 2 adds $2k to the bottom line. I figure I might have some room to negotiate the reconditioning but wasn’t sure if you can negotiate the processing fee.

I plan to bring my own financing and would wait to tell them that until we agree on a final out the door price. For the trade-in, is it any more appealing to a dealer to take the trade in vs if I sell at Carmax?