Good choice on your destination.
There are 2 parts to Michigan the lower peninsula and the (UP) Upper Peninsula inhabited by Yoopers, IMHO the best part. Sorry I had to say that as I grew up there.
You could easily spend your entire time in the UP.
Must see east to west:
1a Mackinaw Bridge and Mackinac Island. Both pronounced "Mac-in-aw". Don't ask me why??!!
1b "The Soo Locks" between Lake Superior and Lake Huron, Sault Ste. Marie MI.
2 Tahquamenon Falls State Park and the Shipwreck Museum in Paradise MI. (Newberry area)
3 The Lake Superior National Lake Shore- Grand Marias to Munising MI. Best seen on the Scenic Boat Cruise on a sunny day.
4 The Yooper Tourist Trap. A one of a kind gift shop in Ishpeming MI.
5 The "Copper Country". The Keweenaw Peninsula and Houghton/Hancock area.
6 The Porcupine Mountains park in Ontonagon Area.
At this point you are only 2 or 300 miles north of Green Bay WI. and the Green Bay Packers Hall of Fame. Might as well include that.
BTW Michigan State Parks (state wide) are some of the nicest anywhere and have very nice RV park areas. Late summer early fall is the PERFECT time to visit the UP. No problem with after Labor Day as the UP is a 12 month playground.
DO NOT miss eating some Pasties (available at most Mom and Pop Cafes) served with gravy or ketchup or both (my personal favorite).
Oops seems your allotted time is gone, sorry.
DN