dome light problems

My car has dome light supervision from the factory. i have installed a Flip Down DVD Player with Dome Lights. My problem is that the Dome light supervision does not work.
There are three wires that was connected to the original dome light

1)Constant 12V
2)Ground
3)Test negative(possibly used by the car's system for door trigger and dome light supervision)

The third wire seem to be what controls the dome light via the car computer(dome light supervision does not work if disconnected) but the dome lights on the DVD Player require a positive to turn them on. I know this can be done using a relay to turn that negative to positive but I want to have the same delay as the other roof light in the car.
What can be done so this will work?

2006 Honda Stream( wiring similar to 2006 Honda Civic)

If you use the relay to invert the signal the delay timing should be relatively close, however the relay will "chatter" as the voltage fades, the dome light delay should function normally at all times, since the processor for this would be before the light itself, so disconnecting it from the light source should have no effect on the fading signal whatsoever.

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COMMIT YOUR WAY TO JEHOVAH AND HE WILL ACT IN YOUR BEHALF. PSALMS 37:5

I agree (inverting relay) but wouldn't expect any chattering since the (grounded) door switch is either ground else floating. (That's taking the relay signal from the diode-OR'd door switches.)