Please don't put a food and water bowl in the carrier with your bunny. If something were to happen like having to stop suddenly, or worst case scenario getting into an accident, it will be even worse for your bunny since the bowls will fly through the air and injure your bunny. Provide water from a small water bottle attached to the front of the carrier (even if they don't usually use one, at least it's there, and try to show your bunny how it works before hand). Definitely put hay in the carrier, just loose at one end is fine. Try to make the surface non slip if you can, a carpet tile covered by a fleece or similar would work. Make sure you shield the carrier from the sun when it's in the car, because it can heat up easily. Also, the thing to be most concerned with is dehydration, offer fresh leafy greens (that your bunny is used to) throughout the drive to aid with hydration. If they won't eat or drink while driving, you have to make frequent rest stops to offer food and water.
If you have the time and haven't done so already, I also suggest researching a new vet near your new house and getting in contact with them to set up your bunny's details. Drives can be stressful and if your bunny needs urgent vet care after you arrive, you want to know where you can go immediately without having to research first.
Hope that helps