Since the OP will have his own car, he does not have to rely on a hotel shuttle if he stays at a hotel near a train station that offers non-resident daily parking. I've stayed at the Holiday Inn in South Plainfield, NJ, and parked for $4 at the Edison train station. I think that the Rahway train station also offers non-resident parking.

The trade-off, of course, is that the further you stay from Manhattan, the longer -- and more expensive -- the train ride will be. But if any of the OP's kids is under 12 and the trip into Manhattan will be on a Saturday or Sunday, those kids will ride for free. And the further south you stay in NJ, the shorter your drive from D.C. will take roundtrip.