With a post this long, I am surprised no one has mentioned Dimple Rapid on the Lower Yough, another dangerous undercut on a very popular river. The decision taken there was to not change the rapid but rather to post large, very visible signs above the rapid telling people to get out and scout and clearly pointing out the danger spot. Seems to have worked though there are still close calls.
From what Tannhauser described, filling the undercut/cave would be a massive effort since he actually underestimated the density of rocks, which is closer to 2 tons/cubic yard for large pieces of rock. In other up to 80 tons of rock could be needed to completely fill a 40 cubic yard void.
From the photos it appears that a possibly better and easier solution would be to block off the right channel near the top where there is boulder splitting the right and center lines. A few foot high stone ledge would prevent boaters from taking the right channel at low water and divert more flow into the center channel making it more runnable. Of course, what this would do the rapid at high water is unknown.
The biggest problem with any efforts to change the rapid is the law of unintended consequences - "fixing" one problem will like lead to new hazards. My vote would be for adequate signage and education. Have the BLM consult with the folks at Ohiopyle if they need to.
"It's All Good"