You just THINK you are a no bleach household.

If you use salt, there may be microscopic bits of bleach ions around. Salt and water are electrolyzed to make bleach. There are electrochemical reactions that constantly happen in the body and in nature.
You can use greensand (contains some form of manganese, IIRC), you may be able to aerate the raw well water enough to tolerate it, but no way would I do either.
Sodium hypochlorite (bleach) is unstable. It loses chlorine to the air just standing. When sodium hypochlorite comes in contact with acids, sunlight, certain metals and poisonous and corrosive gasses it degrades. Aeration will remove any excess through contact with the CO2 in the air.
One of the reasons I use chlorine in our water supply is that it tears apart pesticides and herbicides, converting them to stable inert compounds. Peroxide would do something similar with that, but I don't think it would be as effective against sulfur.
<edit to add: I have a WORKING knowledge. Cabin Fever has a PROFESSIONAL knowledge. While he and I might differ a little, trust his absolute knowledge above mine. If I don't agree, I'll grump at him and try to see where he is coming from.>