the plug needs to be pulled all the way out to energize the brakes.
do yourself a favor and just replace the whole unit. just about every break away switch i have seen on older trailers fails to close the contacts when the lanyard is pulled.
the switch is an inexpensive item, usually around 15 bucks. a new lanyard will be included.
to replace it do just as you suggested, disconnect the battery and change the switch. there are only 2 wires and it does not matter which one gets hooked to the trailer.
also, when you attach your trailer to your tow vehicle do not plug the umbilical cord in until you have attached the lanyard. pulling the plug on the break away may damage your break controller if it is plugged in.
you call them ferrets, i call them weasels.