Originally Posted by
Often1
OP's question is about the checkpoint. The answer is no. BVD's (and a ticket if you are flying WN international) are about WN's internal requirements.
Also, just to clarify, if flying internationally the child (regardless of age) will need a passport, for both Southwest and TSA. TSA may or may not ask for the child's passport at security (depending primarily on whether the gate used is designated for international travel only), but they require all passengers, including children, on international flights have a passport in their possession at all times.