Remember that if you bring this case, it will be your responsibility, that is, you carry the "burden of proof", to show the connection between the babysitter's apartment your son's ingestion of lead. You will need to hire expert witnesses to do that.
The others are correct, that if a dog has shown vicious propensities in the past, you can recover. A good lawyer will help you investigate. Bear in mind that any recovery - whether in a settlement or lawsuit - will require the consent of your workers' compensation insurance carrier, and the carrier will have to be paid back a certain amount for cost of the medical treatment you received.