Having an 19mo and a 3yo is already alot of work, and very expensive. Lets be real, it may be your daughter's hamster but no 3yo is actually caring for a hamster in any real way. You have a sick dog and a lizard that's high maintenance. My husband and I make good money, but we couldn't afford a sick dog (even with pet insurance) never mind a lizard that will eventually need a 100 gallon tank and a hamster who's bedding needs to be changed every other day, double litter box duty AND two toddlers AND 4 days off a week. Perhaps you have more money but seriously--save it.
Adding a puppy dosn't sound like a very good idea. Your 3yo is not going to learn any more responsibility for anything by having a 2nd dog that she isn't already "learning" from a rodent, lizard, sick dog, cats and baby sibiling. Give it a few more years when she reaches 7 or 8 if you want it to help to teach her, but even at that age don't expect much in the way of her actually caring for the animal.