I agree with Andrew: The Bacon Bike Hostel outside of Colville is worth staying at! There is also bike camping between Mazama and Winthrop and the "Bike Yurt" in Republic. Both are on the Warmshowers network, and I believe the bike camping is on the ACA map.

To the OP: Last year we did something like this. To get between Spokane and Colville (where you can connect to SR 20) we used US 395. Most of it wasn't bad: wide shoulder, decent scenery, not a lot of climbs. A decent amount of traffic, though, and it picks up considerably as you close in on Spokane. There looked like ways to get around that, but the alternate routings would have added distance and we were pressed for time.