I am afraid you are out of luck. I live in LA, and I have been to LAX many times.I know that each terminal has its own screening area, so once you are inside, you are stuck inside that Terminal. The exception to this rule is you may travel between terminal 6 and 7 without reclearing security, but as your flight is in terminal 2, there is no way to go from terminal 2 to terminal 6 without going through securtiy. I hope this helps.

Alex