Also, there is no crossing of the Columbia heading towards Ellensburg except the I-90 bridge at Vantage.
That bridge has two lanes on each side, heavy traffic, and zero shoulder.
Where are you getting your route info - Google Maps?
Because there is a lot of garbage in your route - even worse if you have a kid with you.
You can't bike the west side of the Columbia - no roads - certainly nothing but trails.
And the Yakima Firing Range is off limits to entry.
Also, you have some pretty tough gravel roads in the eastern Washington section.
Time of year matters a lot in eastern and central Washington.
If you are going to be riding in mid to late summer - you should consider further north.
The Yakima Valley broils in late summer - plus the plateau is extremely remote.
Heading over the Cascades into Seattle you have 4 options:
WA 20 - over the North Cascades - light to moderate traffic - cool, somewhat remote
US 2 - Stevens Pass - moderate-plus traffic, sometimes heavy - inconsistent shoulders
The Iron Horse Trail - Snoqualmie Tunnel - check for construction planned for next year
WA 410 - Chinook Pass - light to moderate traffic to the east with stunning views of Mount Rushmore
So it seems you want to connect the Trail of the Coeur d'Alenes with the Iron Horse Trail.
Not easy - especially in August.
You can work your way southwest thru the Palouse via St Johns and Lacrosse -
Then via Washtuchna, Kahlotus, and Connell to the Vernita Bridge over the Columbia -
Then to Yakima and up the very scenic Yakima Canyon to Ellensburg.
There is camping at Scootenay west of Connell, rest area w/water at the bridge
And a store where you can get stuff and, maybe, camp behind at the junction of WA 24 & WA 241
I've biked a lot of the back roads of eastern Washington - traffic is light - but can be really remote.