LEDs have a very low current draw and therefore are able to utilize that always-on 2V of power to the dome light. Unfortunately this is not optimal in your situation. This is also why replacing standard flasher bulbs with LEDs usually results in a fast flash. The lower resistance and current draw of the LED differs from what the electrical system was designed for which is why you need aftermarket relays or additional resistors.
I'm not sure what the specification of your LED is but try adding a resistor in between the power source and LED. Perhaps a typical 1k resistor, make sure it's rated for 12V.