You may want to try listing it on Cargurus.com as well.
Unless you find an extremely creative dealership who would pay off your trade and delay the purchase until you close I think you are correct that you will lose the tax benefit of trading in. And I don't even know if that is something they could/would do.
You could swing by Carmax and see what they would offer you to buy your car. They will give you a free 7 day purchase offer.
Just don't apply for a car loan for atleast 24 hours after the house close. Yes lenders do pull credit upto seconds before closing and they to do stop deals at that point.
They want to do final check for dti and make sure you are NOT making in large last minute purchases. I was at a closing where this happened many years ago, Buyer on the way to closing applied for a HD card. Lender pulled credit after that and at the title co when we all got there closer said lender cancelled because of the borrower applying for new credit. Which do want a car only or house and a car? If the anwer is house and car wait 24 hours after closing. If you need a car in the meantime get a 7 day rental.