You can't use the throttle connector.
HWBS has a hall sensor in it so always three wires - 5v, ground and signal. You'd have to wire the HWBS to the two wires in the red connector and the 5v (normally red) wire in the throttle connctor, so cut off all the connectors and solder the wires together. It only takes 5 seconds to cut them and a couple of minutes to resolder them if ever you need to change something in the future, which is less time than it'll take to find matching connectors and solder them in.
Another solution is to use a reed switch and a magnet, which works very well if you have bear cable running down the top of your frame. Use a spoke magnet on the cable so that it'll clamp in place, then stick the reed switch to your frame after figuring out the best position to make the switch work with the brake. You then need to solder the reed switch wires into the controller. One wire goes to any ground and the other goes to the brake signal wire. If you're careful, you can scrape a bit of the insulation off where it comes out of the harness , and splice it on, otherwise solder it to the pcb where the signal wire attaches.
The way the brake switches work is that the signal wire is held at 5v via a resistor in the controller. When a switch shorts it to ground, the voltage is pulled down to zero. the controller is looking for high (5v) or low(0v) on that wire. If it sees 0v, power is cut.
KT controllers have a very fast response time, so you don't really need brake switches. I never had any in 10 years, and didn't miss them.