AA policy gives their lounge access (including qualifying OW elites) to departing AND arriving customers so no need to buy another flight. You can check-in together at any AA LAX lounge showing your KOA-LAX boarding pass.

Re: shower back in Hawaii, in addition to the hotel or beach rinse-off options, also look into resort pass. Maybe you can stop in a spa to relax and shower, and that might even cost less than a hotel night.