Actually Iron(III) is reduced to Iron(II) by hydrogen peroxide. At the same time Iron(II) is oxidized to Iron(III) by it as well. This ping-ponging is the catalytic machinery that makes for the
Fenton reaction. The free radicals produced are extremely potent.
Also Iron(II) is appreciably soluble at neutral or acidic pH. Here is a nice
solubility chart for iron's two oxidation states.
Perhaps if there isn't too much iron then it will cycle between oxidation states faster than iron(III) can precipitate? At least until the hydrogen peroxide runs out, and then the O
2 will eventually bring back the iron(III) hydroxide if your hypochlorite doesn't do it first. So I would expect the brown to come back somewhere unless something else sequesters the iron.
Maybe it just ends up reprecipitating as larger particles of hydroxide and settles. Or maybe it never totally dissolves and the H
2O
2 just sort of accelerates Ostwald ripening?