This one is a mystery to me. I ride a carbon road bike and just recently a clicking noise has developed when I push hard on the left side crank only (right side makes no noise). I have tried different pedals and the result is always the same. I replaced the BB, no change. I have removed, cleaned, regreased all the moving parts. Any suggestions?
A lot of carbon-fiber frames with press-fit bottom brackets have a problem with bottom-bracket creak. This is because press-fit bearings depend on the bearing cups being perfectly parallel and in line, and the frames are fabricated with the bearing cups out of spec.
There are replacement bottom brackets that come with their own sleeves that thread together in the center, which force the bearings into alignment.
It's odd that this would result in one-sided clicking, so this might not explain it.
I just repaired this kind of left-sided click on my bike. The left pedal outer bearing was dead even if not that old. There was no sand inside or anything like that, but the bearing is really grinding even in my fingers and the click is gone with a new piece.
Of course, your case might be different.