I had the same problem. I using LibGDX with Bullet engine, so my code sample on Java, but I'm sure, that it will works on C++ too. Here is my solution (for Z axis):
body.getWorldTransform().getRotation(mRotation);
// That gives you an angle in all range but excluding (85, 95) and (-95, 85). For other axis you can try to get Pitch or Yaw.
float roll = mRotation.getRoll();
// That gives you an angle in range [0, 240). Clockwise and counterclockwise directions isn't detected.
float angle = mRotation.getAngleAround(0, 0, 1);
// Usually 0, but on (85, 95) and (-95, 85) becomes 1 and -1.
int gimbalPole = mRotation.getGimbalPole();
// Using roll (pitch/yaw for other axis) if it's defined, and using angle with gimble pole otherwise.
float rotation = (gimbalPole == 0) ? roll : angle * gimbalPole;
Obtained rotation will be in range (-180, 180). It can be easily converted to [0, 360) range:
if (rotation < 0) rotation += 360;