Since the only highway that actually GOES to the North Cascades NP is 20, I presume you just want to avoid as many of the steep grades as possible getting there. Many of those grades are east of Winthrop and can be avoided.
I've driven all the main roads in that area, and for most forgiving grades here's the route I'd pick:
From Spokane, Hwy 2 to Wilbur, then north on 174/17 through Grand Coulee to Bridgeport. (Do stop at the dam: fantastic!)
At Bridgeport, short dogleg to Pateros, where you pick up Hwy 153 north through Twisp. Connects to Hwy 20 near Winthrop.
Have fun!
