For the DC area defination controversy, maybe one good test is whether the Russians are likely to nuke it directly? If so, it’s part of DC area. 
When was the last time you were in Gaithersburg? I live right across the street from NIST, as in I walk to the end of my street and I see the fencing for NIST. Gaithersburg runs in to Rockville, which runs in to Bethesda, which runs in to DC. It’s all city.
OK. It’s been some time since I was in Gaithersburg, so I’ll defer to your direct experience. Of course, I now live in Metro Manila (population ~13 million), so my concept of a city is something more akin to Trantor or Coruscant.
That area has changed dramatically in the last couple of decades.
Growing up in the area in the 80s, Gaithersburg was the outer rim of The Sprawl in MoCo, but the line between Gaithersburg and Rockville was already pretty muddy.
Germantown was basically just a crossroads with a couple of random housing developments and farmland. Now, where it was once clearly separate from Gaithersburg, Germantown has expanded immensely and they’ve essentially merged, and are threatening to swallow Clarksburg as well.
I don’t know how big it is but the Federal Citizen Information Center is part of the Office of Citizen Services and Innovative Technologies, which, in turn, is part of the General Services Administration. Both of those are headquartered in Washington, DC.
I have been, in fact. 
Friends of mine used to live in Gaithersburg (15 years ago or so), and I visited them there several times (driving past NIST when we’d get off of I-270). And, on edit, I see that DCinDC has already answered the question better than I could. 
I asked a friend who works at the GAO. He suggested the Denali Commission and the Appalachian Regional Commission. The Denali Commission is based in Anchorage but it works solely on Alaska issues so it might be too regional for your purposes. The Appalachian Regional Commission is both regionally focused and based in Washington so it fails on two counts.
He also suggested this sourcebook as another way to generate ideas (PDF warning):
https://www.acus.gov/sites/default/files/documents/Sourcebook%202012%20FINAL_May%202013.pdf
Through that I found the Udall Foundation, which seems to have offices in both Arizona and Washington, DC, but it lists the Arizona office first. Morris Udall was an Arizona representative for 30 years.
Finally, as an amusing aside for people who argue whether Germantown is part of the DC metro area, my friend noted that one reason the Department of Energy decided to put its nuclear offices there was to keep them further away from a nuclear strike. So, at one time at least, the DOE didn’t really think of Germantown as being very close to DC.
They built the Naval Ordnance Lab in the mid-1940s in White Oak, MD (currently the site of the FDA HQ) because at the time it was still very rural and that was considered far enough away from the city and any dense residential areas that their experiments would not disturb anyone. They were still blowing stuff up there in the early 90s, long after the facility had been completely surrounded by residential neighborhoods.
I think any DoD research lab falls afoul of the “no military units” restriction. Even if they’re not a numbered-echelon (like a squadron or a division) organization, they’re usually commanded by a flag officer, not directed by a civilian executive; largely staffed by military members; often sited within a military base (AFRL is within Wright-Patterson AFB, for your instance); and mostly under military discipline and law enforcement regimes.
But of course, I’m just guessing about how seriously the “no military organization” restriction can be taken. The division between military and civilian has been badly blurred by 60 years of military-industrial complex.
I’ve had two quasi-governmental agencies as clients, both of which operate under the oversight of the USDA; both of them are “check-off” agencies which exist to promote particular food products.
One, the Milk Processor Education Program (MilkPEP, the “milk mustache” people), is located in DC. The other, The National Cattlemen’s Beef Association (“Beef – It’s What’s For Dinner”), is located in Centennial, Colorado (though they also have an office in DC).