This is a lot of work, but you can take a shower right at LAX.

Head over to Terminal 4 and buy a refundable ticket on American Airlines. Make it a round trip to lessen your chances for special harassment at security. Tickets to Las Vegas are generally the cheapest refundable ones you can get from LAX on AA. Once you've bought your ticket, have the agent check you in for the Vegas flight and print your boarding pass.

Once you've got your boarding pass in hand, go through security and into the AA Admirals Club. You can buy a day pass for $50. The Admirals Club is much larger and better appointed than the LAX RCC, and has several showers available.

Take your shower, and enjoy the atmosphere in the Admirals Club until it's time to head back to Terminal 6/7/8 for your United flight. Call AA any time and refund your unused ticket to Vegas.

It's too bad the domestic RCCs don't have showers.