SoCal

I know of no airline clubs outside of security. I am pretty certain that TSA would not let you through security into the Delta terminal if you don't have a boarding pass to fly on an airline leaving from that terminal. If you want to be sure, phone Delta.

By "the internatinoal terminal" I assume you mean the Tom Bradley Terminal (several airlines, including United, have international flights from other terminals). The terminal is, I believe, undergoing some renovations, which could affect what lounges are available, but I found references for a Star Alliance lounge and a Cathay Pacific lounge (I think they're separate, since the former is listed as having showers and the latter was not). Air France also listed as having a lounge there. Some airlines have regular club lounges and first class lounges. If you contact Cathay Pacific I'm sure they could give you info, such as day pass rates (unless you're flying international biz class, in which case it should be free). I would use the lounge at your departure, not arrival, terminal so you don't have to guess how long it'll take to get between the two terminals. The free shuttle bus goes counter-clockwise (or, if you want to get some exercise, you could walk).

You could always have lunch or dinner at the Encounter restaurant, in the middle of LAX, if the times work out for you.