I had the exact same problem in my first generation DualShock 4. The controller simply doesn’t work when the touch-pad is connected to the main circuit, if I disconnect it and try to turn on the controller it works just fine. I’ve searched all over the internet and finally found a solution. Just bear in mind that It worked for me but it may never work for you.
Unscrew the four little screws in the back of the controller and remove the cover, unplug the battery and the light-bar wire, unscrew the one screw underneath the battery and pull back the circuit board to get a glimpse of the full spectrum of our problem, it lies in the wire that connects the touch-pad to the main circuit.
Now you just have to figure out which end has the problem but in my case, and much likely in yours, the problem lies at the touch-pad’s end. Remove the cover that’s glued to the back of the touch-pad, if it has any, to unveil the connector. It has a little black string that holds the wire and acts like a lock.
You’ve got to pull it up to release the wire but the problem might be just that, sometimes when we’re disassembling the controller we’ve got to be kinda rough and apply some strength, which forces the touch-pad wire out of its place. Make sure the wire is well connected at the touch-pad’s end with this little black string holding it AND make sure the wire is well connected at the main circuit’s end as well.
Reassemble the controller and try it out, the problem must be fixed.