I think you're using terms to mean different things than they actually mean.
"Nonstop" means a single flight segment from origin to destination.
"Direct" means no change of planes, but can (and often does) involve one or more intermediate airport stops.
"Connection" means a change of planes at an intermediate airport.
"Stopover" normally means interrupting your trip (such as a 2-day stopover in New York City when flying from Kansas City to Paris).
I would gladly pay more for Southwest than for Spirit, even if Spirit is a nonstop and Southwest involves either a stop or a connection at an intermediate airport.
Southwest provides more leg room/knee room, flexibility to change you flight without a change fee, 2 free checked bags per passenger, and a far better reputation for handling problems, such as flight delays.
Do some Google searches and read up on Spirit Airlines.
By the time you add Spirit's many fees (they even charge for water), the prices should be quite close.
By the way, if your destination is
Disneyland, I would strongly recommend Orange County Airport (aka Santa Ana, aka SNA, aka John Wayne Airport) instead of LAX. It's much closer to Disneyland and it's a very nice airport. You don't even have to take a bus to get your rental car.