If you put in at the gate of Button Rock Preserve you can paddle downstream into the Class V- middle section of NSV, which runs into the Class III CR80 run that you are talking about. This is totally legal.
What is not legal is to hike upstream from the gate and run section from Button Rock reservoir down (there is no way to drive to the reservoir by the way, only rangers and private land owners around the reservoir can).
If we can make it though the current 2 year test period and get access for UNSV paddlers granted permanently then we may be able to get them to open up the section below button rock to intermediate paddlers. But that is a battle for another day. They are facing lots of pressure from mtn bikers, equestrians, lake paddlers, etc... to open up the preserve and were lucky as it is to have this 2 year grace period to impress on them that access out of UNSV is not a major liability.