you are going to be in a VERY urbanized area until you get to at least interstate 81 at hagerstown. even it is even very busy with shopping malls for the entire 3 state area. any motel inside metro dc area will cost a fortune and they are few and far between in that area.
there is a really great park at the edge of fairfax county and prince william county it is just off interstate 66 near centerville va, where you can camp for the first night. they have a huge swimming pool and lots of other kids and families for your 3 year old. it is only about 20 miles from burke so not such a risk if the trailer isn't as great as you hope.
one of the biggest rv dealers is located in manassas, just down 66 a few miles if you need anything and have LOTS of money.
who knows,you might even enjoy yourselves, hope so! have a safe trip whatever you do.