The reg doesn't draw power through the brown wire, it just reads the voltage. NO volt drop from the reg reading the voltage.
There are many other places that can cause a voltage drop before the reg reads the voltage on the brown wire.
Starting at the battery, the connection where the cable hook on. A red wire comes off the positive battery cable to feed power to the switch. There is a plug in connector before this wire goes into the harness. One possible bad connection.
Once in the harness that red wire runs up to a plug in the headlight bucket. This plug hooks to the plug on the switch, one possible bad connection. In the switch has a copper plate that gets moved around to make and break connections. Two more places for bad connections.
Back out of switch on a brown wire to the plug. Another possible bad connection.
That brown wire runs to several things but the one we are concerned with now is where it runs to the regulator. There is a plug on the regulator. Another possible weak spot.
These are the places you want to check voltages. These are the places that loose dirty connections cause the low voltage at regulator.
Most often the switch is the culprit. You can take off the switch, Disassemble and clean the contacts in the switch.
Leo