In my opinion, it's not all that essential to check to see if the melty wire is shorted to ground with a meter; obviously it is, or nearly so, since it's clearly been carrying an abnormal current for the generator just sitting there. (One would expect essentially zero current, maybe a couple milliamps at most, when the generator is shut off, and perhaps a couple amps when it's running, and somewhere around maybe 100A when its cranking.)
The solution is to trace along the wire and find where it's shorting to ground. I'd be pretty confident it's somewhere along the wire between the battery and the generator, rather than a problem in the generator, provided there's no obvious evidence of bad things in the generator itself--things like burn marks, or melted insulation on wires, or loose wires on the starter or starter solenoid, etc.
I'm not saying that checking things with a meter is never a useful diagnostic technique, just that in this case the evidence is abundantly enough clear that it's unnecessary. Similarly, if you find a big puddle of oil under your engine, it's probably not too necessary to check the level on the dipstick; you can be pretty sure it'll be low without checking.