First, be really careful having him out where there might be dogs (stray, or just uncontrolled), wild animals, etc. There are even a few diseases that rabbits can get from the ground (possibly wild rabbits?). Pasteurella is the first one that comes to mind, but there are others.
Second, if you are the driver, I don’t recommend having the rabbit in your lap. This is a safety risk, both for you, and especially your rabbit should you get in an accident and the airbag goes off. I will admit that I drive with my dog in my lap, but she has been trained to lay perfectly still during car rides so I know she won’t be a distraction to me. I would never trust my rabbit to do the same, and I am still putting my dog at risk for being killed if the airbag goes off. If he does sit on a non-driver’s lap, make sure that the person holding him has full control at all times so that he doesn’t somehow get by the driver’s feet!
Third, what I do with my bunny, as I do travel a fair amount with her, is I have made her travel crate like a second home for her. She has a litterbox with hay in it, and I give her wet veggies in the place of water. Now that I have done 4-5, four hour long car rides with her between home and school, car travel is routine and comfortable for her. She literally flops in her litterbox and just sleeps the whole time. The crate is large enough that (at 3lbs) she can stand up, and as she is so small, she has no problem stretching out in her box. As you are thinking more about short trips, your bunny obviously doesn’t need this much space, but he should not look cramped in his carrier either. My guess is that he is going to need a medium size dog carrier. Not huge, but a little bigger than the standard ones for cats.