A couple of more possibilities since the dog dish hubcaps rule out a clicking wheel cover.
One. Any chance of a distorted brake rotor shield? A bent shield may run close to a rotor but may only make contact when turning. Normally this produces a scraping sound, but depending on how the shield is bent it's possible for one to click.
Two. Often the subject of reman halfshafts being problematic comes up. Since I've bought and installed what seems like a million of them with no problems I've never bought into that theory very much; at least not 100%.
However, on one of my daughter's cars this HAS been an annoying problem.
Right side fails so I replace it. Two weeks later it goes bad, and I mean BAD in caps. Someone forgot to add the grease to the outer joint at the reman facility. Swapped again and problem solved.
Several months go by and the left side goes. Replace the shaft and 2 months later that shaft goes bad; just not to the degree the other side did. Ordered a shaft and inspection when I went to pick it up showed that reman shaft was no good.
Ordered another reman shaft. Inspection showed this one was absolute junk.
Ordered TWO more shafts this time. One bad and one acceptable at least. Car is back together today and running fine.
Parts house stated their shaft supplier had changed; my advice was dump them and revert back to the original.
My point here is that maybe the original click was a halfshaft and it's being assumed that because the shafts were both replaced then those could not be the problem. Shafts can be inspected by feel when they're out of the car. (I'm making the assumption here the shafts were reman units and not new, unused ones.)
Just more theorizin' and hope it helps.