Took me just over 45 days in Arlington county. They legally have 45 days once you submit the paperwork.
I shoot at Gilbert's small arms http://www.gilbertrange.com/
You have to be a member to shoot there, and I believe the shortest membership they have is a month. I have the annual membership which was around 250$ or so...maybe $300. After paying that there are no range fees. If you bring your own targets and ammo you don't have any additional expense at the range. The annual membership I have gets you a discount on merchandise, and pretty much pays for itself if you buy a lot of stuff.
I believe this will be the closest to you. Other options are Blue Ridge Arsenal in chantilly, the NRA range in Fairfax....