Your problem when its running is the MFI relay coil going "off" and "on." If you changed the relay and its still doing it you need to check the wiring to the relay.
Have you checked battery voltage when running and when not running. My car was doing the same thing when the battery was low, but my battery was almost new. Bad alternator was actually draining the battery. I had more voltage when the car was off.
Also I don't think the MFI relay is a good place to tap into for your hotwire relay trigger. The reason is, the motors fuel and spark is cut if one isn't detected.

