I do like the shuttle option.
I spent my childhood in Washington state. I’ve made the drive between Wisconsin and the west coast numerous times and have been through the Sturgis region a few times, once on a motorcycle. I just never went through GNP. When I rode the motorcycle west I had a route planned for the way back that included the park, but a driver of a pickup truck near the Olympics rearended me on the bike and totaled the bike so the return trip got changed (I had the bike bumper-punched from under me and slid up on the hood and windshield of the truck and then slid off when everything stopped moving and was deposited on the pavement by the bike and walked away with just some internal discomfort and soft tissue stress but nothing broken, over ten years now and I still have a little discomfort at the base of my rib cage, but life goes on).
As for bicycling there are times when the highway is closed to cars, so if I ever do bicycle that road it will be in the car-free period of time. Today I found a couple youtube videos of people cycling the road and it looks like it could be fun.
I can see us parking and doing the shuttle one day, then driving ourselves the next day. That summit hike does sound good. I can easily see spending several days in that general region.