If movement of the vehicle causes make/break in the electrics it really does sound like a problem in a harness somewhere. Totally agree with Sierrafery, (as usual). Does it happen if you rev the engine while stationary? Could it be an engine to body or gearbox earth? Are both plugs to the ECU clean and inserted properly? Is the battery secured against movement? Any obvious wear to harness(es) at P clips or where it/they pass through a bulkhead? All connections to ignition switch sound? Any dodgy bits in harness between ignition switch and fuses/bulkhead?
(Just trying to be logical.)