Some good comments, except ...

Some good comments, except for David Russell.  Chlorine should never be used on stainless steel because it destroys the surface layer that gives stainless steel the protection.  Corrosion is rapid.  Salt water, Hydrochloric acid and Chlorine are all highly corrosive to stainless steels.  

The comment by Steve Russell is correct.  An immediate change that can be made, prior to anything else is to quit using the tablets. We assume you mean Calcium Hypochlorite (about 67%) and not swimming pool chlorine tablets that has other stuff added.   Switch to liquid bleach (sodium hypochloride about 12.5% active) and add the material through a schedule 80 PVC chemical injection quill with a Backflow check valve.  that would add the bleach as water fills the tower for a proper mixing.  (see Neptune, or Koflo Corporation - two of many choices)  PS. We carry these items in stock. 

Karen Baker's comments are correct.   However, the sulfate reducing bacteria are usually hiding out under a biofilm that can hold many different organisms, including Legionella.  The lack of a consistent chlorine feed give the slime bacteria (these are really the true slime molds - the myxomycytes) that once established are resistant to chlorine. I personally have attacked this stuff with over 3000 (yes, three thousand) ppm; and they still come back in two to three weeks. Chlorine just goes over the top of the film and can not reach the polysaccharide glue that holds them on the surface.  So it is best to keep the biofilms off in the first place.  the good news is that the slime molds themselves are usually not classified as toxic, unlike the e coli which is what the regulations are concerned. But the slime molds stink.  

Chlorine is effective on e coli but it lacks the ability to get under biofilms; that is where the corrosion is found that Karen Baker was commenting about.   Over the past 30 years both my group, and others, have developed several formulations of alkylamides that I have used effectively to get under biofilms and break them us to allow other biocides, including chlorine to make the kill.  However, when involved with potable water system: those systems must be totally isolated both before, during and until after the final rinse out.  

The applications on biofilms, and the resultant corrosion concerns, are usually for industrial type of water systems because chlorine additions for potable water systems are usually constant and consistent.

Last week I again found a factory at the end of a city water line with no chlorine in the city water.  That water now as a biofilm problem in eight $5million pieces of equipment.  It probably would not have happened if the city water had maintained an acceptable chlorine residual up to the point of contact with the factory.  I have also found it with residential water systems.