Running a pump in the heater core circuit might work, might not. If the water is picked up from near the thermostat and pushed into the heater return line on the water pump and water circulates in the block - there's nothing to really help draw heat out of the coolant.
With the engine off, and the thermostat likely wide-open it's possible this routing may circulate water backwards in the radiator, but may not circulate much in the block. Won't really do much better than just letting the fan run.
The pump would have to be plumbed to either pick-up or drop-off water at the radiator to get some flow through it. Without adding fittings on the radiator tanks or splicing hoses this is a bit tough - and with it plumbed in parallel, the flow will follow the path of least resistance and not necessarily the path which helps cool things off.
An electric water pump or electric motor driving the water pump in the pits would be the way to go (maybe swap the belt between the electric drive and the mechanical drive between rounds?).
