Gosh, I drive the Beaverton Hillsdale Hwy frequently and think it makes a most unpleasant bike route: swift, heavy traffic, distracted drivers, and off-and-on bike lanes where I think the only safe path is the sidewalk.
Here is how I bike to Beaverton from my home in Hillsdale, with an extension onto Farmington for you. There are no climbs, and you avoid the busiest of roads. Please do be careful at Farmington and Murray!
Please note these transitions:
1. Instead of continuing thru Hillsdale onto the BHH, become a pedestrian (if the traffic is heavy) at Capitol and Sunset, crossing Capitol Hwy in front of the dance studio, southward (left turn) on what would be the Sunset extension toward the Wilson High Schl. parking lot. Jog right behind the dance studio then left onto a parking lot between the Wilson grandstands and Rieke School to SW Vermont. (On some Sundays this is the site of the Hillsdale Farmers Market. On those Sundays, go left in front of the grandstand, thru the Wilson parking lot to Vermont). At the bottom of the Vermont hill, jog left on Bertha and right on Capitol Hill Road.
2. Past mile 1, approaching Multnomah Blvd from Capitol Hill Rd, jog right after the Christian School, thru the phone company parking lot, down to Multnomah Blvd.
3. My drawn path between miles 4 and 5.5 is approximate, indicating a bike/ped path which begins behind the Garden Home Rec Center. An alternate would be Garden Home Rd to Allen.
There are variations to this. Feel free to ask.