Sorry, if any confusion. Twinbrook and White Flint are a couple of stops from Shady Grove (the north end of the Red Line.)
It is my area of familiarity as well, as I live in Frederick County and have worked in Rockville for more than 20 years.
There are several newer hotels near White Flint that are only a couple of blocks from the Metro. Parking is generally free at these hotels. I would probably select White Flint over Rockville. Once you get to Bethedsa, you begin to pay daily parking fees. May or may not be a deal breaker, as the metro fares into D.C. would be slightly lower from Bethesda. Lots of restaurants in Bethesda as well. One other nice thing about Bethesda, is that there is more frequent metro service than White Flint/Twinbrook/Rockville/Shady Grove, in that many of the Red Line north trains will end at Grosvenor station.
Short of driving everywhere on the weekend, I'm not sure how you would avoid Metro. If you are driving to the Zoo, leave early as there is limited parking. Every where else downtown D.C., I would rather (and do) take metro and avoid the hassle of driving and parking in downtown D.C.
MARC is certainly an option, with the limitation being the schedule is only weekdays and pretty much limited to the rush hours.
If I were flying into Reagan National, I would probably look for a hotel near the airport area, maybe Crystal City, in Virginia. Very convenient to downtown D.C., and off two metro lines (blue and yellow).
Personally, if I was going to be in D.C. for only a couple of days, I would look for a hotel downtown D.C. More expensive, but I could come or go as I like. Weekend prices aren't to bad, and once in a while, we will find a deal on Groupon. We will take the old car and park it for free at the Shady Grove Metro garage and take the Metro into D.C. Makes for a fun, and not very expensive weekend.
Jim