I looked it up on yahoo travel and it is about 1500 miles and 24 hours straight through. If you can imagine being strapped down in a seat that, at best, is hard and hot, and you aren't allowed to move for days and days. That is what it will feel like to your child. Time passes much differently for them. I agree somewhat with your husband but again sleeping in the same position, especially sitting up, for hours and hours is an awful lot of stress for anyone.
Is your husband going to stay awake all day and take care of her while you sleep so you can drive all night? No, so I don't think it is fair of him to think you can take care of her all day then stay up all night driving. Either pack your car in the truck/tow it and both go in one car and share the driving or one of you drive and the other take a train with the child. You can get on the train at Ardmore to Dallas/Fort Worth and then get on the train for about 40 hours and he can meet you at the station when you get to New York.
Here's an example of the trip. Leave the Ardmore station, go to D/FW (2 Hr.), change to the Texas Eagle, go to Chicago (23 1/2 hr.), change to the train taking the route you want (20-21 Hrs.), either by the lake shores or through Washington, etc...you can take food and water, sleep in an actual, well sort of, fold down bed.
Or if you wanted to drive in the night time to Newton Kansas you could take the train to Chicago in about 12 hours, then to NY it's only about 20 hours. You might beat him there. LOL.
If you do decide to drive then you must insist that you make frequent stops for eating and play. I would say every 2-4 hours she needs to run and play and have a meal or snack, plus diaper change. Take lots of fluids, that is one thing people just forget to do when traveling and they end up dehydrated. Check out websites for road trips. There are lots of things to do in cars with older kids but not much for little ones except movies on a DVD player.
If it were me I would consider taking along another driver to help out and let you and her sleep through, they could fly back, or you fly there by way of Military discount, take the train, or something to make the trip easier for that little angel.