So with the battery backup, essentially will the battery trickle charger be also providing the power for the pump itself?
You still need the stock power supply. The stock power supply is what the pump runs off when the AC power is working, it also powers the relay in the tunze connecor. It will plug into the female port on the Tunze connector. When power is lost the stock supply, which is pluged into the wall, turns off telling the tunze connector (just a simple relay circut in a nice housling) to switch over to your battery. When power is restored the stock supply comes back on and switches the relay back over to running the pump from the stock power supply.
The trickle charger does not power the pump, it only maintains the battery when AC power is working.