3.8 volts brown wire at alternator ???

[VetteVet]

Are you checking the voltage drop with the brown wire disconnected at the alternator or where ever the end of the wire is. If it is plugged into the alternator, or any current flow ,then you will get a voltage drop anywhere along the wire.

The problem is you're getting way too much voltage drop at the end of the resistance wire.
12 - 3.8 means an 8.2 drop. Or the current draw is way more than it should be.

Measure the resistance on the B/W wire from the key switch with the alternator plug out and see what you have. I would expect 10 ohms which is what the factory rating is.
The charging light in the dash on the light clusters is rated at 10 ohms so the factory matched that with the resistance wire.