Okay, here's "plan b" if you care to try it:
30 - power supply
87a - cathodes
87 - no connection
86 - dome tap #5
85 - dome tap #4
That puts the relay coil in parallel with the dome light...essentially, whatever voltage the dome light gets, the relay coil gets. The only glitch is that the dome light will fade out before the relay shuts off when you close the door.
I had a feeling that the cathodes might end up staying on somehow with the normally-closed contact, but after I re-read the voltages you had measured, I convinced myself that they wouldn't. Sorry about that.
