From the Beach Club your best way back would be via the Disney Studios, as long as the park is still open. If you enter Epcot through the International Gateway you have a very long walk through part of World Showcase and then Future World before you get to the main entrance/exit. You would also need a park admission for Epcot.
You can either walk 20 minutes or so to the Studios or take the Friendship boats that run between the two parks all day. They stop at the Beach/Yacht clubs (closer to the Yacht club), Boardwalk, and Swan/Dolphin along the way.
Your other option is to walk out the main entrance (not boardwalk side) of the Beach Club and take a taxi. Fast and reasonable way to get around WDW. Taxis are normally waiting outside all WDW resorts. If one it not there a Cast Member will get one for you right away.