I don't know if there is an airside bus between Terminals Three and Four. I can understand you not wanting to have to clear security again, however the last time I was at LAX (a month or so ago, the tunnel between Terminal Six (Continental et. al) and Terminal Seven (UAL) had been closed off.

So even if there were an airside bus to get you to Four and you are still able to use the tunnel from Four to Five to Six, (I could) I'm afraid you're out the door and onto the curb if you need to get to Seven.

Perhaps things have changed, if so let us all know.