Why Island Park Idaho for Yellowstone? You are adding a 30 mile one way trip to the West Entrance every day. That's an extra hour on the road each Yellowstone day. Yellowstone is big enough without adding those extra miles. Stay in West Yellowstone if you want the West Entrance.
Fastest route from the Black Hills to West Yellowstone is I-90 to the Belle Fourche exit, Go to Belle Fourche and pick up 212 west (also called the Warrior Trail) take 212 to Garryowen, MT where you pick up I-90. Take I-90 west to US 191 South to West Yellowstone. An alternative is take US 89 off of I-90 to the North Entrance of Yellowstone, that will cut 60 or so miles off the trip to Yellowstone.