No only AD, Retirees, Reservists and their dependents are allowed to fly Space A. Even though your child has one, you do not have a military ID card and listed in DEERS therefore you are not a dependent. You will have to fly commercial airlines to Korea. Until you get married then you can attempt to fly Space A. With the Korea tour you are only allowed to fly 1 time to that location.
Plus with Space A you can be bumped at anytime. You can't fly in a CONUS location, you can only go from CONUS to OCONUS so you would have to pay commercial fare to get somewhere anyway. Flying Space A means any type of plane so it can be difficult with a newborn.
The child will also need a passport have you gotten that yet, the father's signature needs to be on the application form or a Power of Attorney to sign in his name.