I don't think you are understanding how that power supply works. The input to the led's is constant current, in which the voltage will increase/decrease in order to maintain the preset current limit.
I'm thinking that is a generic power supply and the current limiting circuit is in the push button housing, but I could be wrong. It's not in front of me. If it IS in the button housing, there has to be a microchip that regulates the current, and depending upon which one it is could be altered to achieve your goals.
If you can open the switch box and see what's in there, it will determine if it can be altered. You could also just forego all that and get a power supply that meets your criteria right off the bat, so you can just dive into the coding.