I am sure there are connections somehow, but pinning them down is tough. My grandmother, Marie Nolen, was born in 1899, and I remember her speaking frequently about Cole Beamish. Howver, she went to high school in Thompson Falls, I believe, or maybe even in Washington somewhere.
Her uncles, Joe and Frank Boyer, and her mother, Mary Boyer Nelson Scott, lived in Plains, but I believe it was later than the period you are looking for. Joseph Boyer married Mary Lynch, hence the connection. What made it confusing for me is there were now two Mary Boyers in the same town, and I sometimes can't tell who a story is about. I do know that Joe and Mary Lynch Boyer were quite well thought of, and that Mary Lynch Boyer wrote a great deal of Plains early history. There are two books that I know of, "Horse Plains, Montana Territory," by Maurice Helterline; and also "Wild Horse Plains," by John Rhone. I'm sure that Mary Lynch Boyer's historical accounts are somewhere in Plains still, as they were the basis for the Rhone book.
My mother attended the small one-room school house that was moved from somewhere in the woods to the corner of Main Street in Plains, and stands there today, but I don't know if that was Lynch Creek School. My mother was born in 1922, so again, out of your date range. I definitely remember the Beamish name, though. I just looked through my mother's senior almanac, and found that one of the freshmen was named Neptune Lynch. As this was 1940, later than you need. And darnit, for whatever reason, the editors only applied last names to the seniors. All others are listed as Dorothy D., or Mary M. The only reason I knew this was Neptune Lynch was because he signed his name next to his picture, (as Nep Lynch,) which is the nickname I think all of the Neptunes went by.

Sorry I couldn't be of more help, but I will keep an eye out for the names you are looking for as I research mine.
Thanks for getting in touch.