Since he is eating I would offer shreaded or pureed carrots as this often helps making them poo. Carrots have a fair bit of water in them and beardies like the taste so it nicely hydrates them. Daily warm soaks in water is benefical as the massaging the stomach and intestine area, moving towards the vent.
Do you have any knowledge of teh husbantry of the dragon from the previous owner? There is always a chance he could be impacted. Get in touch if possible with the former owner and see if they can say the last time he pooed while in their care.
It also could just be new place stress and the dog probably isn't helping that is keeping him from pooing. Separate the dog for awhile as suggested alrighty till the dragon is accustomed to his new home..could take up to a week or two for this to happen.