ATP for sure. Many majors (including all the highest paid) require it, and all prefer it.
I would only do the 73 type if you want to work for SWA, and meet their minimums. Use the GI Bill to reduce the cost.
Also be aware of this...a 737 type on your pilot certificate with no operational experience sends a very clear message in the airline world:
"SWA is my number one job choice. I'm only talking to your because SWA either hasn't called yet or has already rejected me."
This may reduce your odds a little with other airlines, they will assume that you will leave for SWA as soon as they call you.