Hard to say without knowing the actual current and voltage waveforms.
3.3V and 5V nodes have both a minimum current specification of 0.3 A to keep their voltage limits, I don't even expect that too low load causes a shutdown. It's rather a problem of exceeding the 12V peak current specification.
Considering that the ATX supply has already considerable output capacitance at the 12V nodes, an external 22 mF capacitor must not necessarily absorb the current peaks completely. It depends on the actual wiring and capacitor ESR. A RC or LC filter might help, as suggested.