The monorail does not go to MGM. You could take the monorail from TTC to Epcot, but you would have to walk thru Epcot to the International Gateway entrance/exit to get to the Beach Club. So, essentially, you have to have a park ticket for this way.
You could take a bus to MGM and then take the boat to the Beach club.
Or, you could do the taxi for the quickest and most direct route.