Check and see if that flight is usually full. If it's not, don't purchase a seat, bring the carseat, and ask at the gate if they have a spare seat for a lap infant. If not, gate check the seat. We had excellent luck doing that when the economy was lousy before they cut so many flights; it may be harder now.
Is your child still in an infant carrier? If yes, then installing it in a cab is easy and you have built in transport when you get to the other end. Otherwise, it really depends on where you're going. Airport to lodgings is easy, but sightseeing you can't schlep your seat all day long, and most convertible seats are a pain to install. Some cabs have carseats (in my town, I don't know about DC), but it's not illegal to have her on your lap in that situation.