Instrument cluster voltage regulator

That's what I assumed. I removed the cluster, removed the fuel gauge and the small copper strip in the center you described was corroded badly. Fuzzy green badly. I scraped if off with my pocketknife and polished it up with Brasso and reassembled. It works now. Although whether that was the problem or for how long it will work I don't know.

LOL You must have a memory problem. A nichrome resistor wire in a fuel sender is wound on a phenolic board and has no insulation.
In this application the nichrome resistor wire has a spiral wound multifiber fiberglass insulation that we can hardly see when its new. It is a pale goldish yellow color. Without it the wire shorts to the bimetal beam it's wound on. I have seen this insulation cooked to a black charcoal like. Touch it in a cleaning effort and it crumbles. I've noticed a ground black pepper like residue at the bottom of a gauge and knew right away what I would find when gauge is opened.