There's no reason his enclosure would get too hot unless improperly set up, or you have a heat wave going on and no AC to cool your house down while you're away.
Enclosures should be set in a way that it maintains the proper temperatures of basking (100F-110F), warm side (90F), and cool side (78F) for the entire duration they're on, often 12-14hrs each day. Once my lighting is set up, unless I'm changing a bulb, I do not mess with it. If your tank is getting too hot to the point you're messing with it when home and concerned about being too hot when gone, then it needs to be addressed, likely to a lower watt bulb or raised higher up.
If it is not necessary to bring the dragon along, I would recommend you leave him home. A roadtrip is not always fun for a bearded dragon and would take him away from the lighting he needs and also likely stress him out.
There are many beardies that love roadtrips, but I believe your dragon is still very very young from what I've read, and would likely not take this trip well until he's older and larger.
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IF you do need to take him along, you could use a small critter keeper, a small pet carrier, or smaller 10-20 gallon tank lined with some kind of fleece flooring for the trip. Bring some fresh veggies and some easy to keep and feed live feeders, i.e BSFL or silkworms. Crickets will be difficult to maintain, and honestly the trip might make him not want to eat anyway. Make sure he gets plenty of natural light when you stop, at least 30 minutes but 2 hours is preferable.
