Did the same itinerary a few weeks ago (on a Sunday), no problem. Am a US cit. Through Immigration and Customs and Luggage transfer in maybe 30-40 minutes. Problem was the security line at Term #1 (USAIR and SWA share). As an elite was told I could by-pass the long line which stretched outside the terminal and could use the elevator to the right of the USAIR ticket counter. Bottom line--into the club in less than hour after arrival at LAX.