The Crown Room (where day passes are available for purchase) and Business Elite lounges have separate sets of rules.

If you are a SkyTeam ElitePlus (which includes Platinum Medallion on Delta), then you have access to international lounges regardless of class of service. Ironically, a Platinum cannot use a BE lounge for domestic travel.

Otherwise, if you really want a shower, you can see if an airport hotel offers a short term stay.