On the part of Trans Am I did the following year (between Missoula and Fairplay, CO), I stayed for free in Wisdom, MT, Lander, WY and Walden and Kremling, CO and a couple cheap sites in Yellowstone/Teetons. In Ennis, MT I paid a modest fee at a fish commission campsite. Could have stayed for free in a few more places but I chose not to because money was not a big concern. When you pass through Twin Bridges, MT, make sure to check out the free (donations encouraged) bike camp in town. I was there last year. Great place.
One thing you have going for you is that you are not in a large group. Somewhere like a fire station or church is probably more likely to allow to pitch a tent if it's just you and your son.
Another option is National Forests. With certain limits, you can generally practice what is called "dispersed camping" for free. Note that you generally won't really have access to tap water and bathrooms. Developed sites with running water, picnic tables and tolietes usually cost something.