Several years back, perhaps a decade or more, working on the theory that when there is a click, the case is too big, and/or the chamber too small, in an area that could not be reached by a die, I did an experiment.
I took a fired case that had a pronounced click, slipped a bullet in the neck for support, and chucked the neck up in my cordless drill. Laying the drill on its side, so that the off side of the rim was supported on the laminate desk top (old and well used) I held an inexpensive, coarse diamond lap on the case, so that the edge of the lap was over the extractor groove, and the cutting face was parallel to the CL of the case. With the drill on low speed, and the trigger barely pulled, I reduced the diameter of the part of the case immediately in front of the extractor groove by a couple of thousandths, and then smoothed up the cut with some 0000 steel wool. (Subsequent measurements have shown that this part of the case is part of the solid web.)
On my next trip to the range, I loaded up the case with charges designed to make it click, and fired it several times....no click. Although I did not verify it, I think that this fix would have made it through a match weekend. So there you have it, how to fix a click with the tools at hand. It works.