There are a few places scattered about that rent Harleys and such, but I don't know about dualsport/adventure rentals. On the other hand, the prices I've been quoted for those Harley rentals, well, it wouldn't take many rentals to cost enough to buy your own dualsport or adventure bike. $2K is plenty if you are not trying to impress anyone. Here are some options. How well you shop can make a huge difference in price. Also, I'm assuming you'll ride somewhat conservatively like I do, not assuming every dirt road is your private race track like so many riders do.
The problem with small dualsports is they don't provide much in highway performance and are a real chore to keep up with 80mph traffic. The problem with bigger dualsports and adventure bikes is they don't provide much offroad ability and are a real chore to wrestle through the tight technical stuff, mud, and sand off the highway.
The good thing about smaller dualsports is they are a lot easier to ride, even running errands around town. The good thing about larger dualsports and adventure bikes is they handle the highway with comfort and confidence. With patience you might luck in to a 400-500cc dualsport from a bygone era, like when you could reach the ground while sitting on a wide, cushy seat. Think Honda SL or XL, Yamaha had 400 and 500cc versions, but all the Japanese makers built such bikes as did many of the Europeans. When it comes to compromise for the purposes you state, these are hard to beat. Unfortunately, most seem to have been trashed over the years, and those still available bring a high price.
What it sounds like you really need is a scrambler. Strange as it seems, over the years I've seen many small and midsize street bikes with standard ergos, ranging in age from the 1960s to new, with happy riders. With a set of tires like K270s that provide good performance compromises for dirt and pavement and usually a lower final drive ratio, most any 250-400cc street bike will handle roads ranging from dirt to interstate safely. The Honda CLs from the 1970s are still around, and providing riders exactly the services you specified.
Triumph still makes a Scrambler, but knocking on 500 pounds wet, it is a lot of bike for mud and sand. An experienced rider might can handle it. A newbie will ruin his day, pronto. The Triumph website has a blurb about the common conversions of street bikes to scramblers, and it is still done by many riders because the happened upon a used bike cheap. The other alternative is $9500 for a new Triumph.
Some old wrecks that have been made into scramblers were free for the hauling. Doesn't take much to total a street bike once you add up the labor, and if you go back with cheap aftermarket stuff off the internet instead of OEM, do the labor yourself, and you know you are going to add a few dents and dings to the existing cosmetic damage, you can have a totaled streeter in scrambler mod for $400 or less, easy. That would include new tires, tubes, sprockets and ringed chain, straight handlebars, grips, mirrors, turn signals, another light or two, brush guards, homemade skid plate, and a few other cheap farkles.
There are other options, too, like the Zong Shen RX3, which is my current backroad burner. About $4000 delivered, all it really needs to get started is a brighter H4 headlight bulb. Comes with everything else (like luggage and high enough alternator output for electric gear in winter and extra headlights), though there are precautionary chores before you ride (because it is Chinese). Now that I have some miles on mine, if I had it to do over again I'd check the hubs for square mounts for rotors and sprocket, lower the final drive gearing a tad (I'm spoiled to TW200 torque off idle), and if I'd be hitting the dirt and gravel secondary roads of north Texas, some street legal knobbies replacing the OEM adventure tires. Not a bad deal for your future.
So, there's a passel of own-your-own from practically free to $17,000. Rental near home really doesn't make much sense for most people.