I hiked stevens in very late June or very early July to ski cowboy and skyline bowl(profile pics!), and there are a few dirt roads that go up to grace lakes and hogsback that could be fun biking. Not sure if they would care or not, but I doubt they would care very much..
Oh, theres also a road that goes from bobsled turn(that turn with a barrier in the middle you go by when going up from the western/seattle side, over tunnel creek) to mill valley, and out the nordic trails to the highway on the other side. It'd be a long ride, but you could do just part of it or work extra hard and do it all! (so tunnel creek, switchback up to hollywood bowl, to gemini in mill valley, down to the nordic trails and 5ish miles on those flats..). Haven't seen it in the summer much, but have skied over it a bunch.