I just did the drive by car from Standish (a few miles North of Bay City) all the way up to Mackinaw City. All on Route 23. Very Nice. The road does vary in width (most places with a shoulder) and composition (tar vs concrete) and in condition. Speed limit was mostly 55 if I recall, and not a whole lot of traffic - at least the day I did the drive. But I think it would be enjoyable. There are many state and local parks with access to the lake for swimming, beach, with camping along the way and small towns for food, motel, etc. Much of the route has private homes along the shore so that restricts the lake view and access. Call 888.784.7328 for Michigan travel info. Ask for a free DOT Road map and if you can bike the bridge, but I supect they will not allow it. Call and ask DOT.
I moteled in Mackinaw City. Very touristy, and expensive, especially on the weekend. If you can, avoid the weekend. Motel prices were 50% less weekdays. If you are going across the bridge, I would suggest you see M. City and then stay in St. Ignace in the UP - motels are a lot cheaper but far less to do.
If you have extra time and energy - consider doing a loop from M. City to Cross Village to Harbor Spring and back. The road (route 119) from Cross Village to Harbor Spring is an outstanding bike route. About a 90 mile loop.
Have a good and safe ride.
Robert