I agree.. Day one is going to be brutal. Considering you have metropolis's on both ends you could hit serious traffic.
I think I would look at something like SJU to maybe Bakersfield or Victorville area.
Day 2 Quartzsite to Tucson (unless you have to hit PHX) only because it's an easy run into PHX or Tucson.
Not sure about day 3, but I can assure you that El Paso to Austin is going to be even more brutal. It's as far from Los Angeles to El Paso as it is from El Paso to San Antonio, and you're still 100 miles short of Austin at that point (and that traffic can suck the entire way between the two at times). Even though the speed limit is 80, it's a L O N G way. IMO I would find something maybe 1/2 way from El Paso to SAT, stay there and do a final banzai run up to Austin.
I've done this run in 3 days, but not with kids on board.
PS- I don't know anything about this, but the SIL who was stationed in El Paso for several years says to keep off the border when camping or even hotels. Always trust a local..
Jeff - WA6EQU
'06 Itasca Meridian 34H, CAT C7/350