I've been cycling through Montana this past week. While I've stayed in motels, most of these small towns seem to allow it.

When I came into Roundup yesterday, the local deputy mentioned camping in the park.
Outskirts of Lewistown has a park marked with "overnight stays welcome".
Small towns of Fairfield, Dupuyer, Grass Range all had small parks that looked welcome.

It is also nice to see people mentioning seeing other touring bikes and tourists. I last came through Roundup 17 years ago on my first cross-country trip and somehow seems like I'm hearing more of other cyclists now.