I don't recall a specific case of running the monitor to the battery, but it would be a good idea.
The thing is, the monitor wire shouldn't have current running through it. Voltage, yes, but it's basically just a signal wire to tell the R/R when to crank out current on the white wire or when to throttle back. So ... there's probably a short from the signal wire to ground?
If that were the case, though, those wires should ALSO be heating up when the ignition is on but the engine isn't running, since the battery would be dumping current through there too. Sounds like a bad R/R.
Edit: For what it's worth for troubleshooting, the monitor wire doesn't go through the JB, only the ignition switch.
Last edited by Thorn; 05-23-2020 at 01:18 PM.