You have to address the voltage drop first. Visually check that the main- and starter studs on the starter relay for space between the wire terminals. They are close to each other so the terminals might actually jump a spark and engage the starter for a moment and pull down voltage.
If ok, run a jumper wire from battery + to ignition side of ballast resistor and disconnect the original wire from ballast. If this cures it you have limited the problem down to ignition switch and ignition wiring.

Still no go? Eliminate the starter relay by disconnecting the black(?) Starter solenoid wire from starter relay after starting the car. Test again.

Also check starter terminals for space between solenoid- and battery terminals. They might be close to touching eachother depending on how wires were mounted.

Please check this before you tear into the engine!

Let us hear what you find.

Lars