So to begin with, you can seek reimbursement of your medical bills from Gap.
2ndly, if you want Gap to pay for the baby's pain and suffering, then it is a laiblity issue. Means you have to prove they were negligent. IMO, you have a case here. You should have tried to settle with them in the begining. It is not too late though.
If they refuse to settle, you know what u got to do.