The AC at Dulles is quite small. Since you are on an int'l itin, you should go for the BA club. It's stocked w/ a self service bar and snacks, and it's just a few gates down from where your LAX flight will depart. Since you'll have at least a 3 hour layover at LAX, I would go straight for the Flagship Lounge at LAX after arrival and use the shower facilities there. Then wander over to the Tom Bradley terminal and hang out in the CX 1st lounge.