I am glad it was just a misread on the law..beaded lizards and bearded dragons are completely different animals..and makes more sense they would be banned than bearded dragons.
Though the venom in beaded lizards and Gila monsters isn't that potent but if a place won't allow venomous snakes, beaded lizards and gila monsters, being the only true venomous lizards (or at least considered potentially venomous to humans), it makes more sense.
For traveling, I suggest just placing your dragon in a cardboard box or opaque Rubbermaid container large enough so he cal lay comfortably with his tail curled in a natural position. Line with clean cloth or paper towel. If you plan to leave first thing in the morning, I suggest putting your dragon in his box at night, before you go to bed. This way he just sleeps through the entire trip and wont' be stressed by being put into a box after he has been up and about. Oh also, do not feed him before leaving. You don't want him in a box with fresh food in his gut.
Keep him in temps around 70-80F for the trip (ie don't leave him in the when you stop to have a pee break or lunch. On a sunny day the interior temperature of a car even with windows open, can reach 100F in a very short time). You would be better off putting him into a grocery bag and bringing him inside with you.
I don't suggest taking him out on the trip for 'sunshine' you may end up loosing him that way. The ideal is to keep him in a temporary dormant state to eliminate stress.
Once you get to your new home. Get his cage set up as soon as you can, turn on the lights and then put him inside and leave him be. He probably won't eat right away, so just leave him adjust. You can offer a shadow dish of water. But with such a short trip, once he had a few hours to adjust, you could offer some salad. Or just wait till the next day. One day without food for a healthy adult dragon is perfectly fine.
I am glad to hear you can keep your pet dragon 
-----
PHLdyPayne