There has to be an easy way to build a simple relay timer circuit, probably just as simple as a one or two transistors, couple of resistors and a timing capacitor to hold the relay "in" for a second or two so that you can bridge the gap from start to run.
I would guess that this is originally an issue that was set up in the original ignition switch on the hemi car.
Something like this. Push button would be your ignition switch start contacts, probably fed through a series diode. The contacts would be fed 12V and switched contact would go in parallel with your "run" line.