The only efficient way to get from the Beach Club to the Magic Kingdom would be via bus. There is a direct bus that goes from the Beach Club to the Magic Kingdom. The only other alternatives would likely take as long or longer. Here would be your other choices, but I wouldn't recommend them:
- Walk from the rear entrance to Epcot all the way through the park to the front entrance and take the monorail. I discuss the disadvantages of this
HERE.
- Take a taxi. This is not great, because non-Disney vehicles cannot drive directly to the Magic Kingdom. They must let you off at the Ticket and Transportation Center which is near the Polynesian hotel. From there you would need to take a boat or the monorail.
- Use your own car. Similar situation with the taxi. You'd have to park at the TTC lot, and then transfer to another form of transportation.
From all but the monorail resorts, the Wilderness Lodge, and Fort Wilderness, the bus is the best way to get to the Magic Kingdom.