Depends on two things: the length the axle will span and the load on the flywheel. If the axle spans length is more the full length of the ball (6"+), then use a high strength axle. If your flywheel requires a high amount of compression to transfer energy, use a high strength axle.
I am confused why everyone was so quick to dismiss the idea. For reference, most (if not all) of the 169 teams in TP used HS axles on their world robots. Any minute improvement to the robot is worth it, and a HS axle is simply more durable than a LS one.
So (imo), take the extra effort to get a HS axle to work. In the long run, it will pay off. The last thing you want is right before finals, your flywheel axle bends. The probability might be low, but anything is possible (as history has shown time and time again). Take into account every possibility and work to mitigate it.