Agree with the others about Cherry Hill park. I agree with the others that you do not want to drive downtown. The parking garages are mostly to low and to tight for a RV. I live in the near by suburbs, and before retiring drove downtown to work. There are not many parts of the country that have worse traffic or that have roads as congested.
If you're determined to take your RV into town you might be able to find parking along either the National Mall (access to several Smithsonian Museums) or along Ohio Dr (near the river with access to several monuments) but the area fills fast. Check before you drive because some of the areas are closed to parking for special events (4th of July, National Folk Life Festival for example). I would suggest first visiting via metro and taking the RV the next day. I say this, but still stress you do not want to face DC traffic in an RV.
You need to get there about a half hour before parking opens and be prepared to move a ways when parking enforcement comes by. The parking enforcement is to protect the spaces for the tourist from the people who work near by. Parking opens after rush hour but I have not been down there for a while to see the posted times. (Before retiring I was one of the people that worked near by the mall). The mall and the area along Ohio drive are very nice places to walk a dog, many locals visit the area just for that.
Dog care. Washington DC gets very hot in the summer. In the later part of July and most of August it is not at all unusual to see temps with three digits along with the high humidity. Be careful leaving your dog in any inclosed area. You likely know about inclosed spaces but may not be aware of the summer heat in the DC area. We bought our RV to take our dogs places too. I figure you love your as much a I love mine.