With an extra flimsy cable length, the pump works...?
a 12V mattress pump

These need a surprising amount of amps in order to run properly. No, 3.5A won't be enough. The lighter socket in a car has amps in the 8A-10A range typically.
You try using an old AT or ATX PC PSU, these have &gt;10A on 12V.
Edit: The crocodile clips work as a series resistor that lowers the current enough keepigng the overcurrent protection from triggering inside those wall warts.