Ok I took this random picture of the headlight from the net to explain what I did. I don't have a real picture.
So the line in red is usually where the throttle cable will settle when you have a FZ1 bar. As the bar is lower than the other bars, the throttle cable is too long and will tend to pinch when turn the handlebar to the left. Specially since the FZ1N stay don't have those cable routing loops. that's why when you pull the throttle cable and remove the 'excess' the pinching stops.
What I did on mine and hopefully it will work for you - is to make try to keep the cable along the green line. Doesn't seem like a major change but it made all the difference for me. I used a long zip tie to create a ring (blue) that will stop the throttle cable to go back to the red line. DO NOT tie the throttle cable tightly, leave enough space in the ring so that the throttle cable can move around when you steer the handlebar left and right. Tighten it just enough so that it won't go back to the red position. If you don't have a long zip tie, use two zip ties together. You can attach it to the fork or stay, but not the frame. By doing this, my throttle cable has been free from pinching and everything is smooth. When you put the zip tie in, make sure you test it in all position to make sure there is no pinching before you go riding again.
Of course, if you have too many cables or other things behind the headlight, you might still have this problem. If that's the case, you might want to consider pulling the excess harness inside the frame so that there is no excess behind the headlight. You have to open the tank, remove the air box, battery, battery holder and pull and hide the cable. It was worth the trouble as it freed up so much space behind that headlight. Others have hacked their harness to shorten it which works too, but I prefered not to.
Let us know if that worked for you. If you can't visualise what I did, I could take a picture for you at some point. I will have to remove the headlight in the coming weeks to replace something.