FWIW, I went through this question a few years back and doing research on line (you get what you pay for

I found answers from: yes it will adversely affect the bronze fittings in that sort of concentration to "it shouldn't." I found several examples with pics of chlorine reaction with brass that would indicate it's a problem with brass but I could't find any examples with bronze.
OTOH, all the sewage treatment systems used on boats generate/use chlorine to treat the waste and the sewage usually runs through bronze fittings at some point. I've never heard of a problem and we have those systems on our boat with no issues with fittings. But the amount of chlorine generated/used by the systems is a actually pretty low. I think the concentration of tablets in a strainer could end up being far higher during periods where water is not actually flowing.
In our boat, gasses from the chlorine tablets that are placed in the Galley Maid sewage treatment containers will corrode/break the SS hose clamps that attach the containers to their brackets in a relatively short time - say two years. Of course, that's SS, not bronze. There are no bronze fittings in proximity so I can't tell if there is any affect on them.
Based on what I found at the time, and seeing what the chlorine does to the SS fittings, I decided not to put tablets in any of our strainers but I know of other folks who do it and I have not heard anyone actually report a problem in that specific application.