I think your plan of a couple days in the islands and a couple in the mountains is doable. The islands are all a little different, so you can probably just as easily visit one or several.
I stayed at this glamping place on Orcas Island with my girlfirend last year... it was nice and relaxing.
https://www.stayleanto.com/moran-state-park-glamping/
If you are heading down to Mt Rainier, Mt. St. Helens is also worth visiting.
I would suggest you carefully plan your travel in and out of the city. Avoid peak traffic out of Seattle Friday night and Saturday morning, and returning traffic Monday night (labor day), and you will save several hours of driving.