There's currently a blanket drone prohibition at all national parks. As part of the Orbeso/Nguyen search some drones were used, but the operators were from Grand Canyon National Park and thus part of the NPS "family". They could actually be NPS employees for all I know.
Theoretically it would be possible to apply for a permit to do drone imaging, but I have no idea how amenable JTNP would be to it. My previous relationship with the park's management could best be described as strained, but there have been several regime changes and I don't know what current management's view on it would be. I would never advise anyone to flagrantly disobey rules...that is a purely personal decision. But for me, having worked most of my career in bureaucratic organizations, I value product over process. And Smith Water is a VERY remote place, with some regions being pretty damned dangerous to explore on foot.