[VetteVet]

Based on your limited description of the alternator, and the wiring, I would guess that your problem lies in your using an LED bulb for an indicator. This type bulb only allows current to flow one direction. The indicator bulb uses positive voltage from the key switch and grounds through the alternator diodes until it begins to charge. Once it starts charging it provides positive voltage to backfeed the light and it goes out.

The indicator needs to be a regular incandescent bulb. It's duty is to excite the alternator through the internal regulator to charge the alternator fields to begin charging.
I suspect that the current is going to the alternator full time causing it to overcharge which would account for the warm condition. You can check this with a voltmeter on the output wire of the alternator.

The wire off the ignition switch is a resistance wire equivalent to 10 ohms so it was used as a backup for the bulb in the charging light dashes and when the factory went to all gauges they kept the wire in place as the resistor they needed for the alternator. It must have some resistance to the back current generated by the alternator or it might cause the engine to continue running when the key is shut off.

If you are using a newer CS style alternator, then 10 ohms is not sufficient to protect the alternator diodes and more resistance is needed. The range varies depending on who you listen to but 35 ohms 5 watts to 300 ohms and 1 watt is what I see the most.
Perhaps you can post some pictures of your project and give us the year to help us help you.