Natural Choices makes a good oxygen-based bleach product called Oxy-Boost Oxygen Bleach. (Please note that not all of their products are non-detergent.)
To contact Natural Choices Home Safe Products 1-866-OXY-BOOST and www.oxyboost.com .
I have no connection with this company, but I use this product and like it a lot. My only caveat is to be careful when using this product with very dark blue fabrics. I experienced color transfers to other fabrics that didn’t happen without this product. I expect this isn’t much of a problem since most people would never use any kind of bleach with dark blue, but if you do, simply don’t mix the blues with anything else – the product doesn’t seem to fade the colors much, it just seems to sometimes facilitate transfer of some blue to other fabrics if the load is mixed.