We were in Nashville this summer, not with the camper though.  We did not rent a car, just used Lyft to get around.  We stayed a bit south of Belmont University and a Lyft ride to downtown was around $10-11 and we never had an issue getting one.  Our most expensive ride that weekend was from Broadway out to the Opry, it was about $17.  It was cheaper to get "home" that night, probably because it was late at night and not during peak hours.
So, not the cheapest option but cheaper than paying for downtown parking, IMO.  And of course cheaper than a rental car.