I have lived in the area for years. Played every course near here numerous times. My thoughts.
Home Course. Great property, always in good shape. But every time I play it I leave thinking how much better it could be. It is less than 15 minutes from Chambers.
Eagles Pride, formerly Ft. Lewis. Open to the public, it is not restricted to military. Wonderful course, good mix of old holes and some newer. A classic northwest course with huge trees. Not always in great shape but very challenging. Pretty much next door to The Home Course.
Hawks Prairie. A thirty six hole complex through houses. Excellent courses, the Woods was intended as a private course, but did not materialize. Probably 30 minutes from Chambers.
Gold Mountain. As previously mentioned, really good John Harbottle design. Very northwest, big trees some steep climbs, very fun to play, and always in good shape. Has a sister course that is never mentioned but is very good as well. It seems far, but the drive over there from Chambers is not much more than thirty minutes.
Someone mentioned Washington National. John Fought designed it, and it is really good. It is close in quality to Gold Mtn. But to get there from Chambers is a hike through very unpredictable traffic.
If I were you, I would try to get on Fircrest Golf Club. It is only a few minutes from Chambers, and is terrific. AV Macan did the original design, GCA member Jeff Mingay has been heading a restoration project there. Chambers Bay, Fircrest would be a fantastic one two punch, and would fit your timeline perfectly. Enjoy your visit.
There are some other good spots, but not in the same class as these.