How long to clear international arrival Terminal B LAX and transfer to Terminal 1?

You're proposing to book two separate tickets, as Southwest does not have interline partners. At a minimum, I would book the later flight, and I would give serious consideration to not doing this at all and booking it all on one ticket.

With two separate tickets, you are responsible for your own connection. You must go through immigration, collect any checked bags, go through customs, change terminals, wait in line at the check-in/bag drop counter, check in your checked bags before the deadline, go through security, and get to your gate. If your inbound international flight is late or you're stuck in line at immigration, Southwest is not responsible for your connection. They may, as an unofficial policy if you notify them ahead of time, put you on standby for the next flight, but otherwise would be in their rights to charge you for new tickets on the spot. And with 3 kids, there may not be room on the next flight for your entire party, so you could be waiting longer.

The terminals (terminal 4 and terminal 1) are not close either. There's a inter-terminal shuttle bus every 10 minutes or so, or it's about a half mile walk. The good news is that at that hour, immigration wait times aren't long, generally not more than 20-30 minutes.

The best approach, especially traveling with kids, would be to book this as a single ticket. This will make the airline responsible for your connection, will avoid the need to carry baggage half way around the airport, and will reduce some of the risk and uncertainty.

If you do proceed with separate tickets, the 9:00 is a nonstarter, particularly if you have checked bags and three kids to get to the other side of the airport. The 11:00 should be doable as long as you aren't significantly delayed anywhere and are ok assuming the risk of separate tickets.

Go with the 11:00am flight, especially if these are two different tickets.

If things are normal, there is an ok chance that you can make the 9 am flight but there isn't a lot of room for error and in general LAX is fairly unpredictable. Travelling with 3 kids (as I have done a lot) adds to your unpredictability so extra 2 hours seems like a good safety measure.

You can check statistics on immigration and customs clearance time at https://awt.cbp.gov/ and try to to draw your own conclusions. Keep in mind that you may have to check in again at the Southwest counter and go through security again.

I would go with the 11:00. The walk from Terminal B to 1 can be about 20min at an adult pace so with kids it may talk a little longer. Luggage carts are free in the international terminal, and that might make your trip easier. The security at Southwest can also take a bit of time. I've done the LAX to Southwest a few times, and having 3 hours or more really reduces the stress level.