Before you can make a claim against the cosmetic company you would have to first obtain the records of your visit or visits to them. Once you have the records you would need to have a reputable cosmetic surgeon or dermatologist review the records to see if the treatment fell below the standard of care. If it was below the s/c then you have a valid claim for damages.
the question is whether or not the medication prescription fell below the standard of care for a physician in your community. You need to obtain an expert opinion from an OB/GYN before knowing whether or not to proceed. In addition if your baby does not have any residual injury from the alleged malpractice there may be insufficient damages to proceed. You should check with a malpractice attorney in your town.