The only issue I see with this logic is that the timeline doesn't support your plan. If you get to the gate and haven't purchased a ticket, either (1) the flight will be full and you'll have to have the child on your lap since there are no seats left to purchase or (2) the flight has seats and you don't need to purchase a ticket for the car seat to be in an empty seat. In some instances with a long standby list, the agent may or may not know if there are seats available until the end of the boarding process. Just something to consider.

Also, you may consider purchasing an infant fare, which can only be done over the phone - 1 (800) 435-9792. If you haven't seen the Baby On Board page online, I recommend reviewing it.

SM disclaimer: The opinions above are my own and not necessarily those of Southwest Airlines.