The parking there is not meant for theme park parking and there are signs as you enter stating this fact. So they will give you a 3 hour parking pass that is enough time to eat and check out the resort.
The parking there is meant for guests staying at or enjoying the amenities of the resort.
I would park at the TTC and monorail to Chef Mickey's or just take the bus to MK and walk or take the monorail to Chef Mickey's.
Only other possible option is to valet park and hope that they don't notice that you've passed the 3 hour mark.