I'm not sure about either of those hotels specifically, but I know that a lot of the parking garages in DC that do valet have their own parking system where they line the cars up in long rows, which means they would need your keys.
I had the same dilemma when I signed up for monthly parking at the closest garage to my place, which just happens to do valet. In the end I told them I couldn't leave my keys because they close at 7, I only have one set of keys, and I want access to my car in case of am emergency. So they essentially gave me a spot that's out of the way for them and that I use to park myself. Maybe you could try that approach, though I doubt the hotel garage closes at all.