As I write this, I know I will be labelled a crazy Canadian (or worse), but we took our 31' 1974 Sovereign
to Florida and back, some 4,000+ miles, with our 2007 Ford Freestar van.
I'n with you: I do NOT want to drive a truck the 88% of the time I drive the vehicle with no trailer attached.
My trailer is some 500 lbs heavier than yours, casitaplata, and although we didn't tour the Sierra Nevadas, we did go through lots of hilly country in Kentucky and (if I remember correctly) Tennessee with no trouble.
I also agree with the suggestion of the GMC Acadia, and would also suggest that the new Ford Flex would also do what you want.
However, although they are all good cars, none of them gets great gas mileage. Better than the Dodge 2500, I'd imagine, but I didn't really look up the numbers.
A Freestar would be the cheapest bet, since you won't find a newer one than a 2007. Because they were discontinued, a nice, late-year, low-mileage 2007 loaded up can be had quite cheaply.
The other two might easily stretch your stated budget.
Good luck with your search. Remember that the key
is getting the right hitch set up professionally and properly.