Do the test one more time. If it's the same result, you'll need a new controller. Look on the controller's label t see what the maximum current is. I can't remember what the Big Bear has. I seem to remember 20A, though it might be a bit less. Look at the other controller and compare the maximum current. It might be lower, which will work, but you get a bit lkess power. If you want to try the other controller, make sure that each connector has the wires in the correct sequence to match. If the other bike has a different LCD, it still might not work unless you use the matching LCD.. There are ways around that, so do the tests, try any changes and report back the results.
MOSFETs are like electronic switches. The motor is connected directly to the battery, but the MOSFETs are switched off. the controller opens them in high frequency timed pulses to work the motor. When a MOSFET is damaged. it doesn't switch properly and the motor can't work. Your green one looks OK, so is working and tries to turn the motor, which is what causes the click, but the others might be compromised and prevent the motor from turning. It needs all three to work properly. Actually, there are 6 of them - two for each colour, of which one works for a forward pulse and the other a backwards one. you need both forward and backwards pulses on each colour to make the motor go forwards.
