Is it legal to takeoff in a crosswind that exceeds the design limit of my aircraft?
The 15kt crosswind for the 182 is not a design limit at all. If it were a design limit, it would be specifically stated in the POH. To be honest, I think there's no such thing as a design limit for crosswind. It all comes down to how much rudder authority there's left at a specific crosswind strength. And even that depends on many factors like approach speed, flap settings, weight, CG, etc.

This is what the POH for the C182T says:


  Demonstrated Crosswind Velocity is the velocity of the crosswind
  component for which adequate control of the airplane during takeoff
  and landing was actually demonstrated during certification tests. The
  value shown is not considered to be limiting.


and:


  The maximum allowable crosswind velocity is dependent upon pilot
  capability as well as airplane limitations. Operation in direct
  crosswinds of 15 knots has been demonstrated


So it is perfectly legal to land and takeoff with a stronger crosswind component. It's up to the pilot to determine if it's safe to do so or not. It all depends what the pilot's capabilities are.