Auxiliary water pump

Hello all,
I've been at the track a couple of times this year for the first time. What I notice is that the engine takes forever to cool down after a couple hits on the 1/4 mile.
I don't want to install an electric water pump but would like to plumb in an auxiliary water pump to move water only when the engine isn't running. So after a lap, I shut it down, flip on a switch for the electric radiator fan and a switch for a water pump.
I don't want to cut into the top or bottom radiator and install the pump there since it wont be running when the engine is and I don't want to restrict mechanical flow.
Ideally, if there is a way to push water through the engine using ports on the intake and ports on the water pump that would be great, but I just don't understand how that flow works and that if it will push water through the block/heads and radiator doing it this way.
Any suggestions?

motor water flow.jpg