I'm not sure what you mean by 'breakfast bus.' There will be no direct bus transportation from the All Stars to the Beach Club or back from the Beach Club to the All Stars resort. The buses @ your resort will run to DTD and the parks, starting about 1 hr. before the park opens.
The closest park to the Beach club would be DHS &, @ least for Aug., DHS opens @ 9 am on Sat.s, which means the buses won't start going in that direction until 8 am, I believe. Note, I am NOT an early riser, so someone who makes rope drop will have a much better idea of how early buses start running. Also from DHS you'd have to boat/walk to Beach Club to reach Cape May. And after breakfast, boat/walk back to DHS, and grab the All Stars bus @ the front of DHS to go back to catch ME.
Your safest bet is to grab a taxi to and from breakfast, not sure how much that costs from the All Stars.