Hi Ben,
You are in for a treat!
Mickey's Backyard BBQ is such a good time. This picnic-style meal in an open-air pavilion comes with lots of fun and unique entertainment. My daughter had a blast dancing with Chip 'n Dale at this event!
Mickey's Backyard BBQ is located in the Settlement area of
Disney's Fort Wilderness Resort & Campground. If it works with your schedule, I absolutely recommend that you travel from Magic Kingdom. In order to get from one resort to another, you would need to transfer transportation at some point. My suggestion on getting to Mickey's Backyard BBQ from Disney's Art of Animation Resort would be to head to Magic Kingdom to make this transfer, so if you are already at Magic Kingdom, you are halfway there!
The really good news is that to get from Magic Kingdom to Disney's Fort Wilderness Resort & Campground, you can take a boat. The boat docks at this Settlement area, and you will be very near to Mickey's Backyard BBQ! If you arrive at Disney's Fort Wilderness Resort & Campground by bus, as you would arriving from any other theme park or from Disney Springs, you would then need to take the Fort Wilderness internal bus system to get back to the Settlement area. The internal bus system is very good; however, it is another step.
From Magic Kingdom, I would allow yourself about 40 minutes for this journey, in case you need to wait for a boat, and then there is about a 5 minute walk to the location once you arrive. If you left from Disney's Art of Animation Resort, I would allow closer to 90 minutes in case you needed to wait for the bus to Magic Kingdom.
I hope that these thoughts are helpful for you, Ben! Please do let us know if you have any further questions.