Washington DC

You are better off taking metro down rather than driving your class c. It can be done, but it isn't easy. If you really feel you need to, your best bet is to arrive before 9AM and do it on a Saturday or Sunday. Parking rules are quirky and strict. There are locations that you can park for more than two hours but they are very hard to find without local knowledge. You might also look for a surface lot rather than a garage, but it will be a long walk to the mall from the closest ones. If your dog is not a service dog, it cannot go inside the buildings so visiting early in the morning, around 7AM, is best. The heat and humidity are both low and the area is much less crowded. If the dog is a service dog, I suggest taking the metro down so you don't have to worry about parking.

Best of luck and enjoy your trip!