I say exchange it.
A new battery that does not work... that is simply wrong. After all, maybe it was discharged or maybe it is defective. If I am not mistaken, full discharge is not good for standard batteries (that is what deep cycle batteries are designed to do, but not regular car batteries).
The trunk tool kit can do the job, but you might bring a 13mm and 10mm socket (if I am not mistaken: battery terminals 10mm & hold-down 13mm?) to remove and replace the battery at the store. Assuming that you will not get stranded and the battery is the total cause of failure.