HD 1979 fatboy fuel petcock valve leaking

It's just a stuck float valve on one of your carbs. They need a bench clean. And get the cables routed exactly the same way as they were.

To see what is wrong you will have to drain the tank and take off the fuel tank outlet(filter assembly). What you should see is a tube sticking up with a fuel filter over it. The tube allows the fuel to enter the carburetor down to that level , when the fuel petcock is in the ON position. When you switch to RESERVE the rest of the fuel flows down through a hole at the base of the fuel petcock. If the tube is missing from the fuel tank outlet(filter assembly) or/and the actual fuel petcock switch (sometimes a separate part )you will not have RESERVE. Another way you have no RESERVE is if you have a leaking four hole washer in your fuel selector. This allows the fuel to flow from RESERVE to ON inside the selector valve(gas leaking in petcock but not escaping outside petcock). "Honda" has all sizes of 4 hole washers available, but , the assembly may not be designed to be taken apart and reassembled. In either case you may end up having to buy an assembly to fix your problem.

Your petcock is vacuum operated. Gasoline leaking from the vent hole indicates that the vacuum diaphram is ruptured. A new petcock assembly is needed.

Sounds to me like a cracked fuel line. No leaks when sitting still but add the torque of the engine turning over and the vibration and the line flexes enough to leak. If not a fuel line, the weld may be cracked where the petcock mounts to the tank or the petcock itself may be cracked. Is the petcock water trap screwed on tightly?