Palouse to Cascades would be fine until Kittitas plus abut 2. From there east the conditions a bit more than a month ago was mixed soft sand that was too much for my CX bike and 33's. There is a great alternate route to Vantage and then down the river to the restored crossing at Beverly. For an added vista and a bit of extra climbing riding the PtC until just over the restored Renslow Trestle is worth it and good surface. From Beverly east there are a number of paved or good gravel roads and routes.
Chinook Pass on the other hand is a beautiful route. Extremely Narrow, with no shoulders and cliff on the side you will ride on. Depending on the winter we have there might still be lots of snow up there. I am guessing but mid-week that you are planning is likely the safest. Weekends will have much higher traffic.
Seems like a fun trip.
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Sir Mark, Knight of Sufferlandria