+1 to what Spockster said. The stator will measure as shorted if it is plugged in to the R/R, but should measure open line if unplugged.
You were right about the white and white/brown wires ... the white is voltage from the battery to the ignition switch, and the brown/white powers some accessories, but is also the monitor wire to the R/R, so there's definitely a short somewhere either in the R/R or the stator. You should get an open line reading from the positive lead on the R/R to the negative lead on the R/R. Likewise, there should be an open line reading from the positive lead on the R/R to all of the stator leads.
Unfortunately, the brown wire has little to do with those leads, and shouldn't really be carrying any current, just a voltage the circuit uses to see if the bike is getting the right voltage out of the charging system. If that wire is melting, there's likely a short in the the thyristors of the regulator causing an uncontrolled battery dump.
If you're willing to sacrifice another fuse, unplug the stator from the R/R, but leave the R/R plugged in to the bike's harness and turn the ignition on. If it blows, you're for sure looking at a bad R/R (and the stator should still be inspected, since they tend to die in pairs). If the R/R alone doesn't blow the fuse, you're looking at a bad stator.
Chances seem pretty high that your R/R has failed. If you have to replace, it's highly recommended to get a mosfet replacement instead of the OEM shunt style.