A certified Reptile vet would be a very good idea, as if you see an experienced one they should prescribed him a prescription-strength antiparasitic rather than the over-the-counter remedies that not only don't typically work very well or at all, but can also contain cheaper ingredients that can be harmful to your beardie. Think of it in terms of a dog with ticks or fleas. You go to Walmart and buy a cheap Hartz or Pet Armor flea and tick spray or spot-on and it doesn't kill the eggs left behind, only kills the live ticks and fleas crawling on him. Then you take the dog to the vet and they give them the good stuff that kills the eggs. And you end up spending twice as much as you would have had you just taken him to the vet to begin with.
I'd also be very careful with any of those things you put in their enclosures that are supposed to keep them free of bugs, they are literally poisons that you're hanging in their enclosures and it's always there, constantly exposing them to the poison. I know lots of people who have lost their pet birds because their bird got mites or feather lice and they went out and bought one of those things and hung it in their cage, and the exposure to the poison killed their parrots.
Safest way is an experienced reptile vet giving you a prescription that will kill both the living mites that are on him (you cannot see them all and can never see the eggs) and the eggs they lay on him as well. You need to sanitize EVERYTHING in his enclosure and the enclosure itself, but I wouldn't do it until you get him treated because you're just going to reinfect the enclosure and the stuff inside it. As stated, to kill them all AND the eggs, soapy water won't do it, you need to soak it all in bleach water, rinse well, then bake in the oven AFTER you get your beardie treated, or keep him in a another enclosure until he gets treated. They're hard to get rid of because of the eggs, so the vet is so much easier.