I remember that road being long and dusty, but don't recall any nasty spots, but it has been awhile, so I may be forgetting something. That said, for work, we frequently rent vehicles, sometimes for lengthy periods of time, and use them heavily, including frequent use on easier 4WD roads. We sign the normal contract that prohibits use on dirt roads and we've never had any trouble 'getting away with it'. Here's what we do.
For starters, don't flash your climbing gear and don't mention anything about the mountains. Just act like a dumb tourist, say you're going to Yellowstone or something, or better yet, just visiting people. When you're done, just make sure to wash the car real well before returning it (including under the hood, the wheel wells, and the undercarriage). The rental car agencies tend to check for stupid obvious things (e.g. dirt, dents, scratches, etc.) but almost never think to look at important things (i.e. the undercarriage).
Even if you do bang up something underneath the car, they probably won't even notice, but this won't happen on the road you mentioned, as long as you drive slowly. If you scratch up the side of the car, just put a cheap coat of turtle wax on the car, it will be nice and shiny, and nobody will be the wiser, but this won't happen on the way to the Sandy Lake TH either, because there is no vegetation to scratch it up.
Finally, if they say anything, just deny it. They can't prove that you left the pavement, especially if there's no damage done. If you're still concerned, pay a little extra and rent an SUV and you'll be fine. Oh yeah, and if you get a flat, just pay the $20 to patch it up and don't even mention anything to the rental agency.