Your best bet would be the Washington State Fairgrounds in Puyallup - any month
except September (that's when the fair is going on and they chase out anybody except vendors). So that one's out.
We stayed in the Seattle KOA and hated it. (Bad area, very few sites that would accommodate your rig and that's where our bikes were stolen last year). Did I mention that for all that, it's like $60 a day? Ugh.
The city of Auburn runs a really nice park, $25 a night water/electric. Plenty of sites that would accommodate your rig. Might give them a call. Realize we're slowly moving away from Seattle at this point. (
Website link)
Those are the only three we've stayed in around that area.
Almost forgot - we've stayed in a campground over in Gig Harbor. That would involve paying a toll every time you went back over the Tacoma Narrows Bridge towards Seattle or Tacoma ($6 for the truck, $12 for truck and trailer - it is a one-way toll coming in, no toll going out). Depends on how much you might be driving back and forth that direction.