Hi Peter ~
It certainly sounds like you have an action packed day planned! I am very used to traveling with a son in a larger special needs stroller. Since you are parking a car at the Magic Kingdom handicapped parking lot and would need to return to get it at the end of the day, I would drive.
When you leave the Magic Kingdom and go to Chef Mickey's, you can walk to Disney's Contemporary Resort (it is an easy 10 - 15 minute walk) or take the resort monorail from Magic Kingdom the one stop to the Contemporary Resort. From there you can take the resort monorail back to the Transportation and Ticket Center (TTC) to your car.
You can then drive to Epcot then onto Animal Kingdom. If you're staying at a Walt Disney World Resort hotel, parking is complimentary at all four of the theme parks. If you're staying off property, you will only need to pay for parking once and can then visit as many theme parks as you would like that day without being charged again. At the end of the day, you will have your car with you and won't need to return to the Magic Kingdom parking lot to get it when everyone is tired and ready to head back to your resort.
If you would like, from the Contemporary Resort, you could take the monorail to the TTC and then hop on the Epcot monorail. It is about a 10 - 12 minute ride and it is fun to view the resort this way. You can take a bus from Epcot to Animal Kingdom. If you leave your car at the Magic Kingdom lot, you will need to take a bus back to the Magic Kingdom or one of the resorts along the monorail line and then get on the monorail back to TTC to get back to your car.
If you're staying on property, you may want to take the bus from your resort to the Magic Kingdom and start from there. Also look into the
Express Transportation option to go to park to park.
Wishing you minnie magical memories :-) Allison V.