I have worked with Hydrogen Peroxide in factories in the olden days. By the time the cotton gets through the supply chain I don't see how any could be left on dry cotton. That is WHY they were bragging about using it instead of chlorine. It's not a "persistent poison" like some chemicals, it just burns.
You might enjoy this: Even 15% Hydrogen Peroxide can make paper or cotton catch fire as it dries. But the caustic stuff we bleach hair with is 3% and if using it to rinse my mouth or clean my ears, I always dilute it 50%, so that means 1.5 %. And I spit it out.
But the 15% stuff, if poured on paper or cotton then thrown away in a trash can, will start a fire. Because it heats up a little bit from the conversion from H2O2 to H2O + O2, then it's saturated with oxygen like in a room where someone is in an oxygen tent. Well, things that are at all thin and flammable will catch fire if you look at them cross-eyed in a high-oxygen environment.
I have trouble picturing anybody putting cotton into a plastic package while the peroxide is still present in any amount that could harm anybody, because I'd think the bag would get too much oxygen and create a shipping danger.
From Wikipedia: By volume, dry air contains 78.09% nitrogen, 20.95% oxygen
Here's a paper on the toxicity of DRINKING A WHOLE GLASS OF industrial-strength peroxide, the guy was mostly OK after ER treatment, but ONLY because he realized his mistake and followed 250 ML of H2O2 with 500ml of water.
But that much inside your body at once can create oxygen bubbles in your bloodstream that can kill you, because gas bubbles in your blood are not good. The heart is not designed to pump air.
Accidental ingestion of 35% hydrogen peroxide
I use the unbleached cheesecloth if available just to be "green" but I would not be even slightly afraid of dry cheesecloth or string originally bleached in H2O2.