Not to scare you, but if the infestation is that bad, I'm surprised they haven't attacked the chams yet. I would do whatever you can to keep them protected(which I'm sure you are). If you are seeing droppings everyday, all over and chew holes through stuff, the infestation has probably gotten far beyond your control. Rats aren't just smart, they're intelligent. They have their own distinctive personalities and the ability to learn quickly. Much smarter than reptiles. Which makes them great pets and terrible pests.
You probably know all this, not trying to be a downer. Just putting it out there. Wish the best for you and your family, I know you got a little one, I can't imagine how stressful this would be.