I expect from your results that this controller is setup so that it requires the PAS pulses in order to work at all. It may also respond to throttle *while PAS is going*, or it might be setup to only operate from PAS until you change the mode from the LCD controls (that you don't haffve).
As for using a Cycle Analyst, you can always use one with any controller...it just can't communicate with the controller like the LCD the cotnroller is designed to use does. It is a completely external control unit using just the controller's throttle and brake lines to perform feedback into the controller, and using either the controller's own shunt or an external shunt to monitor battery current.
There is no way to use any CA to change any settings in the controller or make it behave differently in regards to whether PAS or throttle work or don't work, etc. It also can't read any data coming from the controller. For that you'd need either the LCD or some other device with the right communications protocol and "language" and data order, etc.
The PAS, if it's designed to take pulses to keep it operating, could probably use a 555-timer chip based circuit to make continuous pulses on it as along as power is turned on. This would fool the controller into thinking it was always getting the PAS-active signal, fi that's all it uses it for, and then use the throttle to control the actual speed/etc.
If it actually uses the PAS to control the speed/power/etc, you could still make a throttle to control it, if the PAS still uses pulses. You'd use the same 555-timer circuit, except one of the fixed resistors (that determines frequency) would be replaced with a potentiometer that is your throttle, directly or indirectly. Then as you change the pot value with your throttle, it'll change the frequency of the 555 output, which then fools the controller into thinking the PAS is spinning faster or slower, which then makes it spin the mtoor faster or slower.