You're much better off just flying to South Padre Island
I am not convinced that is a slam dunk. The travel and arrival time seems about the same based on doing some "Expedia" and cost could possibly be cheaper to go to Mexico. The Southwest flights seem to be better than what is on the travel sites. I think this will really depend on the type of flight you can actually get. But from a quick look I found on average
LAX to Cabo (La Paz would be a much shorter drive, but did not see any direct flights)
380.00 @ 2.5 hours flight time + 2.5 hours drive, flight arrival around 2:00 pm
In water by 5:00pm
LAX to Harlingen
500.00 @ 5 hours flight time + 40 minutes drive, flight arrival around 3:00 pm
In water by 4:00 pm
The 6am out of LAX to Harlingen gets you there early around (11:00 am), but was close to 580.00.