The black is a conductive protective coating I believe. The copper is the conductor. There is nothing wrong with the copper runs that I can see. All is still connected. The connector just lifts off with prying with several plastic tools in two or three places. Gently raising it up. Not that big of a deal. Slow and easy.
You mis diagnosed your fuel gauge problem. Nothing wrong with the PC board except flaking coating. All of your gauges and bulbs "have wires going to them" in the form of a copper conductor.
The copper does not need cleaning except around contact areas at gauges and bulbs. Cleaning the runs does nothing and may remove copper.
I think you may be missing a thin plastic sheet covering. Not sure.
An option is to take the caps off of the fuel gauge and take the nuts off and the gauge will come loose. If you think its the copper runs or coating that is your fuel gauge problem, after testing the gauge or getting a new gauge or sender, put gauge on again and wire it separately with wires and crimp on ring tongue terminals attached to the gauge terminal studs.
Chances are your gauge problem is the sender, the gauge or ground. Testing needs to be done. You are getting ready to damage the PC board, entire gauge cluster and there is no need to cut connector wires.
