Does anyone have problems with hubcaps rotating?

Tweek the mounting tangs just enough to make the side of the tang in the direction of travel dig in a little deeper, sometimes that's enough. If that doesn't work, screw a pair of small self tapping screws into matching holes you've drilled in the rim on either side of one the mounting tangs, a dab of sealant on the threads wouldn't hurt. Mounting tangs squeezed into a rim to hold a wheelcover on certainly aren't going to jump over screw heads.
While I don't have that problem now because I'm running cop car wheels that look good powder coated white with chrome lugnuts, I've had it in the past with similar stock full covers on a 14" wheel.