I think adding a fairly large capacitor (about the size of a grape) to the output of the power supply will help it stop responding to the sudden changes in the load current. Maybe 2.2 to 3.3 ohms in series with the supply voltage, then a 220 uf/10v aluminum electrolytic capacitor. It might work with no resistor at all.
ps, have you measured the actual current load? What is the amperage rating of the power supply? I don't expect these to be the problem, but verifying that you actually have what you expect is always a good thing to do.