the little critters are treated like princes and they ought to be thankful they are not out there in the mountains dodging birds of prey and snakes.
The thing is, these guys never
were out in the mountains dodging snakes. These guys are so domesticated, they could not survive out in the mountains in the Andes, so comparing the two is like comparing apples to oranges.
A bath now and then probably won't do horrible damage, but it would be better if you would blow dry them to be sure you get them thoroughly dry to prevent fungus or a respiratory tract infection. A blower dry set on cool and on low will take a little while, but it will ensure that they get dry all the way down to the skin and avoid problems later.
Why don't you try bathing them in corn starch for a while and see what happens? It very possibly could be the dust, and the water bathing gets rid of all the residue from the dust, thereby making the itching stop. I would also consider wearing long sleeves and taking a Benadryl when handling them, rather than continually bathing them, which at some point is might end making a mess of their fur and all dry up the oils on their skin, possibly causing
them to itch constantly.
Since I brought my animals into my home, and they didn't choose to be here, I do my very best to accommodate what is best for them, rather than what is convenient for me. You just may need to do a little experimenting to see if you can get around whatever it is that is bothering you. Maybe try liners instead of bedding with a litter box in the corner, which may help if you truly believe it's the urine causing the problem. Try the Benadryl before handling them, if you're going to be handling them for long periods of time. If you get a rash just transporting them from their cage to their play area, maybe try transporting them in their dust bath while wearing some kind of protective covering.