It depends on whether or not the pad turns on the LEDs the same way the TE pcb does, so it could be possible. In order to find out, you’ll need the multimeter.
With the stick unplugged, you first need to see if the LEDs share a common line. You might be able to eyeball a trace that goes to one half of each LED, but if you can’t see one, then you need to use the multimeter set to check resistance. You’re looking for one half of each LED that is all connected together (low to zero resistance). If you find one, draw a quick picture so you can find it again in a second. If you can’t then it wont work, stop here.
Once you’ve found the common LED line, you need to know what the voltage of it is when the stick is plugged into a 360. Plug it into the 360 and turn it on so that one of the LEDs is lit. Take the multimeter, set it to read voltage (the lowest voltage setting that is greater than 5 volts). Put the black probe on Ground somewhere, and the red probe on one of the common LED points. Read the voltage on the meter. If the voltage is low, say less than 1 volt, then its backwards from what the TE LED board uses. Stop. If the voltage is somewhat high, somewhere over 1 volt, then you’re golden.
If it checks out, solder the ribbon from the USBVCC on the LED board to the common LED spot you tested. For the four LED lines on the ribbon cable, connect each one to the NON-common half of each LED on the pad. Test it out.