The monitor should be monitoring amount used or resistance pulled. By powering that relay you might be pulling just enough extra to cause it to think they are out. Or it sees the lows on.
Try this..Disconnect 86 High beam wire from the relay. Turn the lights on and off to reset the lamp monitor. Now power 86 with a test lead from the postive side of the battery and see if the monitor acts up again. If it does, then it is seeing the low beams and you might need to install diodes in to the line where it reads them being on.
If not, then it'* the amount of power the relay is drawing and I would need to think further how you can power a relay or something w/o drawing enough power to upset the monitor. (probably a little circuit w/caps etc to up the voltage)
For those that don't have the lamp monitor. We hate your monitorless guts..