I am not a pro.
I would consider re-sanitizing at the cistern, and use the bleach and vinegar. I think I would recirculate the water from the hot (turned off) back into the cistern, and make sure the recirculating water has sufficient chlorine and low-enough pH to make it effective. I would not aim for the 200 ppm chlorine I aim at for the well, but I would like to see something like 25 or so recirculating. Don't run 25 ppm thru the softener for long.
Long term, H2O2 (hydrogen peroxide) injection may be good. People seem to have good luck on H2S with that. Typically you inject the H2O2, and iron precipitates. Ideally, the precipation would be into a contact tank with a blow-down port to let you blow out the sediment (rust). Then typically the H2O2 gets removed with GAC (granulated activated carbon). But some residual, maybe 5 to 10 ppm H2O2 continuing on thru the pipes and WH.
https://terrylove.com/forums/index....izing-extra-attention-to-4-inch-casing.65845/ is my sanitizing writeup where I discuss pH, etc. For the well I use a flooding volume. For the cistern that would not apply.
I am not sure what would be best for you, but I would start accumulating the vinegar, get chlorine strips, and get a pH meter along with the packets to make the calibration solutions with distilled water.