Google Maps is not necessarily your friend.
It is notoriously inaccurate - especially for cyclists.
Not sure of your experience level, how you tour, and how much of North American you have done already.
Three weeks is not a whole lot of time and less so if you add flying in and back.
Also, what time of year would you be considering?
I have cycled B.C. from top to bottom, Alaska, and the Yukon many times.
There are remote, dirt forest roads where you are little more than bear food.
And there are busy highways packed with cars and trucks zooming by.
Others are right about Juneau - there is no direct land connection.
The Yellowhead Hwy - Route 16 - between Prince Rupert and Prince George is pleasant -
but nothing to write home to Mom about - esp. if you have other untried options.
If you haven't ridden the Icefields Parkway in the Canadian Rockies - it is superb.
One possibility would be to fly into Calgary - ride the Parkway up to Jasper -
Then ride down to Kamloops, BC via Hwy 16 & Hwy 5 along the Blue River.
That would be about 500 miles / 800 km.
If you cover more miles, you can bike into Vancouver.
Or if you spend more time enjoying the parks, you can catch the train back.
I would certainly spend one day in VCV at Stanley Park and the world-class museum there.
Pic - Mount Robson - with incredible backcountry camping you can bike into