It depends on what your bench power supply does when the current is exceeded. If the bench supply shuts off then the pump will stop. If the bench supply drops voltage the pump will slow down.
It is probable that the peak current is starting the pump motor so it may not start at all or start slower depending on the conditions above. You might be able to charge up a high farad capacitor array to supply the peak initial motor start current, get the PSU past that hump till it can supply nearer the needed current.
Personally I would just buy an AC mains powered pump instead, or get a 12VDC supply capable of more current, for example a cheap generic ATX PSU, the kind of cheap low quality product that you'd never want powering a PC, should do fine with intermittent duty powering an air pump, but since it is a cheap low quality unit, you'll probably need to put a dummy load on the 5V rail to keep the 12V rail at a high enough voltage.
Also consider that a good design for a 12V pump would allow it to also work at the higher voltages a vehicle electrical system can generate, so you would have better pump performance shooting for say 13.8V instead of 12.0V, and if the peak is 10A at 12V, it would be higher at 13.8V.
If you just want to use the bench supply and it's shutting off, put a low ohm resistor in series with the pump. It'll run slower but it will run.