The only ground in the gauge system is at the tank sending unit, which could be at the unit itself, or the tank itself may not be grounded good due to some rust etc. (?)
There is no connection between the tail light wiring and the fuel gauge wiring, in the diagram, so it's hard to understand how the lights affect the gauge. So somehow, the grounds have to be the culprit. The grounds for the tail lights are easily seen when removing the side marker light assemblies. Each side has it's own ground behind the marker light, which is easy to remove for a look. The screw in those grounds can be replaced with a stainless steel sheet metal screws in place of the originasl, to insure a good connection.
And on the left inner side of the rear quarter panel under the car, is the plug which connects all the rear lights etc to the front harness (5 pins in that connector) Which is also easy to get at, to check it out.