I'm going to be flying the same route in 2 weeks and will be in the same boat as you.
I've done it a few times before though, and generally always walk it from T4 to TBIT, which are right next door. I never thought about doing airside transfer.
I looked it up on the OW website and they say the following:
http://www.oneworld.com/ow/airports-...ap/los-angeles
"The airport shuttle bus runs every 7-10 minutes and takes 45 minutes from Terminal 4 to TBIT and 10 minutes from TBIT to Terminal 4."
So there is a bus, but it doesn't say if it's airside and I wouldn't want to trade a 2 minute walk for a 45 minute bus ride!