The car battery test is a good one, but make sure you include an appropriate fuse. If there is a fault, the much higher current available from the battery could fry the motor and invalidate the warranty even if the pump was faulty from new, as the makers stipulate a 5A fuse.
Alternatively, just measure the voltage at the pump leads with a DC voltmeter - 12V +/- 1V would suggest a pump / pipework problem, anything less suggests the supply is inadequate. This is quite possible - the normal running current is shown as 3.5A but this is an average figure. A single-diaphragm pump such as this has a pulsating load that peaks once per stroke, that might peak over 5A especially when starting (although not long enough to affect a 5A fuse). Starting from rest could be tripping the supply into 'foldback' immediately, where the voltage drops away to reduce the current to a low value, until the load is removed to trigger it to reset.
I would not have chosen a power supply intended for LEDs, I would be looking more at a 10A / 120W general purpose supply. Note that an electronic power supply behaves quite differently from a traditional wirewound transformer-rectifier when faced with an awkward load. A transformer will ride it, and just droop a little, eventually overheating if heavy overload continues for too long. An electronic one is more likely to go into some kind of protective mode even on a very brief overload such as starting a motor.
If memory serves right, these pumps make about 120 strokes per minute (don't quote me on that). From the rate and the sound one can usually tell whether a diaphragm pump is stiff, or the pipework obstructed, or the supply is faulty. Or maybe I've spent too much of my life lying in bilge listening to pumps and generators and things.