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What is the specific gravity of $\ce{H2O2}$ bleached viscose? In literature, I can find that this is $1.52~\mathrm{g~cm^{-3}}$, but for bleached viscose is it not findable. Can I presume $\ce{H2O2}$ treated viscose has the same density?

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You can assume that the density remains unchanged.

Hydrogen peroxide simply oxidizes dyes, thereby destroying chromophores (groups that move the absorption of a molecule into the visible spectrum). The viscose structure should remain largely unscathed, and the change in density is so minuscule I would argue it is within the experimental error.

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