Can I buy a seat for my < 2 year old infant, but not bring on a car seat?

I would be holding the baby the entire flight (essentially as a lap child) but we want the extra seat so that we have extra space and don’t have to bother our neighbors.

I ask because I read that southwest is one of the few airlines that won’t let you buy a second seat for yourself. And I wasn’t sure if there were any rules specific to infants without car seats.