Without knowing the exact specification of the Zig Mk3 I cannot tell you if you need a battery or not.. The reason is two fold, firstly depending on the design of the Zig unit, it may not produce a 'smooth' 12V supply, it could be a half or full wave rectifier output which is effectively switching off 50 or 100 times a second. which may not suit the pump or its control circuit. Secondly the Zig unit by itself may not have sufficient current capability to start the pump. In either case the system would need a battery to smooth the current and to provide the starting current.
If on the other hand the zig unit is a smoothed power supply, then it should have enough current to turn the pump, so the problem must lie elsewhere.
Sorry not much real help, but hopefully a better understanding.