If you have a car, I would most definitely drive.
But Disney often takes into account the early opening times of restaurants within parks and/or within resorts close to them. However, keep in mind that MK might open at 8 AM that day, meaning you could catch a bus there as early as 7 AM. I'd look for the earliest opening park that day and take a bus there, then from there to AKL. Unfortunately, except for driving, buses are the only transportation going to AKL. Keep in mind there will probably be a morning EMH at one of the parks, meaning buses would run even earlier to that park.
If you have a car, driving would be much faster, simpler and more direct. You're allowed to park there for 3 hours. Then you could drive back to SSR or to the park you wanted to go to (parking is free at the parks for resort guests).