I think maybe you bought the wrong switch. The part you bought has an LED for illumination but no current limiting resistor. So if the amp doesn't have a current limiting resistor then I would expect the LED to burn out pretty quickly-which it sounds like it did.

I don't know what Outlaw does to drive that light. The switch you bought is available with 12V AC, 110V AC, 12V DC and various other illumination options. If you have a volt meter you can measure the voltage across the terminals on the switch. Once you know what that is you can order to proper switch.