You can use any 36v battery you want. There's two wires to the controller, which are black and red. When you connect your new battery, you only have to get them the right way round.
If that were my bike, I'd chuck that whole monstrosity on the back of your bike and replace it with a bottle battery. Your controller sits in a box at the front of the battery, so you would need to find a new place to house it. I put mine in a toolbag under the saddle.
The control system for your bike is very out of date. Modern ones have many more features and give a much better riding experience. If it were my bike, I'd be thinking about getting a bottle battery with an integrated controller, which will make the installation a lot neater and more convenient.
Is your motor sensorless (3 wires) os sensored (3 thick an 5 thin wires)?