I'm going to guess that it is coming from the drain, not from the faucet. There is probably a bacteria culture established in the soap scum residue. Pouring bleach probably would knock it back for a while but it is likely to return unless the soap scum is cleaned out.
I had the problem in my shower drain and was pouring some bleach in it every week. I finally got rid of it by running a brush down it. I used one of those 3 inch brushes made to get dust out of a dryer vent.
I have also had bacteria get established in the vinyl hose of the hand sprayer at the kitchen sink. I took it off and ran a wad of cloth with bleach through it like one would do cleaning a gun. My cold supply line to my bathroom vanity also had the same issue which was resolved the same way. I keep my hot water temp high enough that bacteria cannot establish on the hot supply. The bacteria never did come back, so I'm guessing I contaminated it during installation and that there is not a bacteria problem in my water.