Problems with LED dome light

I highly doubt the LED was the cause to your battery dying.
LED's use VERY little energy. I too have an LED dome light and it does the same thing, a very dim light when it's supposed to be off. I suggest you look elsewhere for your battery dying. Is it the original battery? It may just be time to get a new battery, good luck.

Battery was AAA and within three year warranty so they replaced it and commented it was likely a "phantom drain".

All the vehicles that I have installed the LED dome lights in have stayed slightly lit, There was an explanation about it on another site but I forget what it was. Its fine though. As far as your AAA battery, every single car that comes in with one of those batteries test bad, IMO they are junk.