The switch in that light kit works perfectly on any cable brake. It has two wires and the switch goes on when you operate the brake just like any other brake switch.
Other easy options for a brake switch are:
The sensor off a cycle computer. It also has two wires. Screw the magnet to somewhere where the inner cable is exposed and position the sensor nearby so that the magnet moves over the sensor when you operate the brake. the best position is on the crossbar id you have exposed inner cables running along it. You have to experiment a bit to adjust the magnet into the position for the correct switching point.
Hidden wire brake sensor (HWBS). https://bmsbattery.com/ebike-parts/459-hwbs-hidden-wire-brake-sensor-1pcs-parts.html
Modern controllers have the three-wire connector. If you have two-wire connectors, the yellow goes to ground (black) and blue to the signal wire (not black), then you need to join the red wire to any other 5v red, like the PAS or throttle red. Most of the 5v reds are joined to the controller in a bunch. Often there's a spare pad where they solder. so you can add a wire if you don't want to splice into the throttle or PAS.
Other options are any normally closed reed switch (some have three wires and can me NC or NO) and a magnet. Fix them to a brake lever so that the magnet comes to the switch when you operate the brake. Not as safe, but you can use a normally open reed switch and move the magnet away from it when you operate the brake.
Lastly, you can use a hall sensor and a magnet, which works the same as a reed switch, but you need a 5v supply wire and probably a pull-down resistor to ensure good switching.