Hi Paul
You need to think about how to power this pump.
It would be tricky on a belt drive Whirlpool because AFAIK there is no terminal on the timer that corresponds exactly to when you need the pump powered up and when you need it to be off. (You don't want it draining during the wash for example.)
During the drain/spin part of the cycle, the main motor runs during the initial pump out (no solenoids engaged), until the tub is empty, then the spin solenoid engages. But you can't connect the electric pump across the main motor terminals, because then it will pump out during wash.
One solution that occurs to me is to connect the pump in parallel with the main motor (as I described above) but ALSO wire the electric pump through a change-over relay (with NO - Normally Open and NC - Normally Closed contacts) and connect the pump through the NC contacts, and connect the coil of the relay to the wash (agitate) solenoid - what this will do is power the pump whenever the main motor is running, EXCEPT when the agitate solenoid is also energized. You would need to find a relay with 120V coil if you are in USA or Canada. This should make the pump operate for the neutral drain and the spin.
Of course this is getting quite unnecessarily complicated, you are probably better off just replacing the pump with a new one of the original type, if you can find one.