Jim -
Here’s a couple other things to keep in mind:
The distances between places in MT can take a day of travel getting from one place to another.
In a normal year, you contend with runoff on many streams until early-mid July, and in dry years (such as the recent past half dozen) you’ve got to be cognizant of forest fires, and/or low, warm water temperatures in the latter part of the summer through to early fall that can close down the fishing in some places.
Stream access will be more limited on many of the larger rivers (such as the Clark Fork, already recomended) if you do not have a boat. If you decide to fish one of these rivers, you may want to consider hiring a guide.
Here’s a couple other ideas:
You’ll probably drive by the Big Horn River in SE MT (at Hardin) going and/or returning to MI. A day floating the river with a guide out of Ft. Smith would be fun, and you can tent within a stone’s throw of the river at the Cottonwood Campground 3 miles below Ft. Smith.
A couple hours farther west, the Yellowstone River, upstream or down from Livingston is always a good option. Again, if it were me without a boat, I’d consider a guide for my first time on the river. However, Yellowstone’s Edge RV Park is one of the best campgrounds in MT and is located right on the bank of the Yellowstone about 20 miles south of town. They have shaded tent sites so close to the river that you could cast to the river from your tent if you wanted, and they have private, easy walking access to over a mile of some of the best fishing in the river from the RV park. (There’s also a couple resident fly fisherman who live in the park all summer and fall that might take you fishing for free, including going to some off-the-beaten-path locations (well, maybe if you buy their gas).
The Missouri River near Wolf Creek (just below Holter Dam) and Craig, MT can have excellent summertime fishing, and has easy access, although it can really crowded during mid-summer. There are several places along this stretch of river where you can pitch a tent at minimal cost, or just sleep in your vehicle overnight for free. There are many places here where you can park right beside the river and get out and fish.
There are many other places that you could consider, a lot of which would depend, of course, on the type of places you like to fish - big rivers, small streams, lakes, etc.
John