The walk from the Beach Club to Epcot (International Gateway) is 5-10 minutes tops. Unfortunately they do not let you past the rope until the "official" opening time. For EE this would be 8am. The problem is that the main entrance is closer to Future World so these people always get to the main attractions like Test Track first. We did EE two summers ago and by the time we go to TT the line was already building fast. We grabbed FP and then got on stand by. The wait wasn't too long and because of a ride problem they let us stay on after the first ride. When we got off we used our Fast Passes and got on again right away. I would guess that in about 30 minutes we were able to ride three times.
If you really want to ride TT and MS you probably better use EE. Test Track often runs out of Fast Passes early in the afternoon. Of course, it depends on what time of year you are going.
Mission: Space was not opened that summer so we left Epcot after riding TT and went over to the Studios.
In most situations it's better to use EE to get on major rides and then leave for another (less crowded) park. We have used this strategy with great success.
oh boy...plus trying to get the rest of the family to move quickly will be a task in itself.
We'll were spending the afternoon the day we get there at Epcot. Hoping we'll get lucky with TT & MS. It's the kids #1 choice for 1st rides. If not we'll try during the week.