I've done the road lottery several times. You can do it in a regular car if needed...the road is actually in good shape, and the speed limit is a strictly enforced 35 so there's only so much trouble you can really get in to. That said, with all the flooding in Kantishna this year it may be pretty rough past the Eielson visitor center.
Every time I've done Road Lottery I've seen rental cars out there. The bar codes make them stick out like a sore thumb. I would say tread carefully, and get an SUV to be on the safe side. Bear in mind, too, that it's not unheard of for Denali to see snow that time of year. I've seen it myself at Road Lottery.