Not sure if there's something similar in TBIT, but when arriving int'l via T4, there's a side pathway pass customs for connecting fights allowing for entry back into the terminal. This will save a considerable amount of time, and will allow you to try the KE lounge or PF Chang's (not sure if they still participate), if you're so inclined. The KE lounge will generally have some chicken wings, pizza, some sandwiches and Remy Martin XO at around 5 pm. I generally stop by there for a quick shower before leaving or a connecting flight and for the XO afterwards.

If you chose this path, there's a shuttle that will take you to T1, but I've found it easier just to walk. Doesn't take long. Rock's will likely not too much of an issue during lunch, but just remember to place your name in the queue and do the PP thing for the print out receipt, which will save you some time when you close your tab. They have a pretty cool Bloody Mary with the strip of bacon, which is what I'll usually order.

The AS Lounge, while underwhelming as observed above, often has the sign out, but they're actually accepting guests. On a few occasions, I've seen the sign only to enter and see there's hardly anyone there, and the attendant seemed puzzled about the sign.

PS - in my experience, just say you're using the lounge and you can access any part of LAX with whatever BP you have.