In their typical 737? The overheads never looked big enough for a bass. Maybe a guitar.The person you spoke to at Southwest was wrong.
Just flew LAX to Dallas via Southwest for some video shoots: bass went in the overhead bin both ways at no charge.
I’ve flown with an instrument likely over 100 times in my career, and the only time I was ever charged an “oversized” fee I was flying American: the agent at the check-in counter on the way back home was just being dumb (I even explained I hadn’t been charged on the outbound flight) and I was also using an ENKI case (which is exponentially larger than a normal hardshell case).
That charge was later refunded.