On BMSBattery's Q-series line and on some Bafang hub motors,the white wire is a speed sensor output. Often can be ignored, but is often useful for some controllers.
It can be hard to find controllers with your specific PAS, brakes, LCD connectors. Those are desireable for commercially sold ebikes, but rarely seen on the kit conversion market. You may get lucky. You would either have to buy the PAS, brakes, etc that come with the new controller, or splice your existing ones to the crappy connectors that come on many controllers. Can you get a replacement from the place that sold you the bike? WOrth paying a little extra. Not worth paying 2X-5X extra though.
You do have to find a controller compatible with your old display too, or buy a display with the new controller.