Have you read Proficient Motorcycling: The Ultimate Guide to Riding Well by David Hough? I don't think all the info on counter steering would help much but perhaps riding within sightline, lane positioning and delaying apex might help.
Just curious if his techniques could help or if your KLR's naps are more of a matter of statistics with with the amount of riding you seem to do.
I know those roads out of Brevard. Some of the best in the country except speed limit is kind of low near slippery rock. I like a parallel dirt road (perfect for the KLR) next to 215. I think it is called headwaters road / NF475B and Government Rd 475. At least the knobbies are off the pavement! Of course I ride it a bit slower due to the gravel and not wanting to crash when no one else is around. Still it is quite interesting / challenging even at the lower speeds I ride.