If someone can shed some additional light on how to solve this (no pun intended or is there?), I would appreciate it.
Hi Paul, the heart of the matter comes down to the type of load each device places on a circuit. An incandescent lamp is a purely restive load. LEDs are well, diodes. They behave very differently when it comes to placing them in circuits (one is AC, the other is DC at the element level, for starters). Since your light was designed for incandescent lamps, the switching circuitry in the unit may not completely cut the voltage to each lamp when it is supposed to be off. A triac control circuit would be an example of this, whereas a relay control circuit should be a complete "off". Adding a
dummy load in parallel with an LED light source can sometimes fix this issue when connecting things to dimmers, but it's still dicey as to whether or not it'll work. It's also considered bad practice, among other things. Since your light is an integrated unit, I don't think this is a practical idea here since the load needs to be downstream of the control circuity.
Dimming LED bulbs is a completely different can of worms despite having a similar end result.
Honestly, either use your unit as it was intended or go buy what you need - there's really no sense in trying to modify it. They make LED versions of this exact same effect that can be had for probably less than you spent on your 4 LED bulbs...certainly if you buy more LED bulbs to experiment! Good luck!