What time is your ADR? Even though the signs at the SSR bus stops say the buses don't start running until 45 minutes before park opening, they actually start very early for Magic Kingdom. This is specifically because of the early morning breakfasts and tours. The buses to Magic Kingdom start running before 7:00 am. If you tell the driver you're eating at Chef Mickey's, they'll usually stop and let you out at the Contemporary. If not, it's still only a 10 minute walk. I don't think you need to pay for a taxi.
We got to our 8:00 ADR at 1900 Park Fare at about 7:30, and that involved taking the monorail all the way around to Grand Floridian. It was a surprise to us to find out we'd gotten there before the restaurant even opened.