I think your estimates for doing it yourself are a little high. 3,000 miles at 10 mpg equals 300 gallons of fuel at less than $2/gal is under $600. Assuming 10 days travel (avg 300 miles per day so as not to kill yourself) with 5 nights in RV parks (and 5 in parking lots) at $30/nt equals $150. May take a little searching to avg $30/nt. And if you prepared most of your meals in the RV you would not spend much more than you would not traveling. All total should be less than $1000.
You could travel some long days and mix in a non-travel day or two to rest and still make the trip in 10 days.
