You don’t generally want to move a lot of weight with the triceps kickback. If the weight is pulling your upper arm down, it’s probably too heavy for this move.
That said, make sure you don’t swing all the way in the opposite direction. Yes, you want to be using a light weight, but experiment until you find one that is both light and challenges you during this movement.
If it’s tricky to find a suitable weight or your gym doesn’t have the precisely correct weight available, use one weight increment down from what you think you need. Then, use a high rep scheme and emphasize slow eccentric movement with an especially strong squeeze at the top of each rep.