Fastest is by car or taxi.
If you prefer Disney transportation, take a bus to DHS and walk or ride the boat to the Beach Club. You should allow at least an hour to do this. The boat ride itself will be about 15-20 minutes not including the time you may have to wait for the boat to arrive. I also believe the bus you take from ASM to DHS may stop at Blizzard Beach unless it is closed. Do not take the bus to Epcot unless you have park admission available to walk through the park. You must enter the park to get to the Epcot resorts from the Epcot bus stops. If you end up at the Epcot bus stops and don't have park admission you would probably need to take a bus from Epcot to DHS and go to the Beach Club from there.
If you factor in possible transportation delays and the time it will take you to check in once you arrive at ASM, I think you should plan to take a taxi.