You don't just sterilize fabric with bleach. (How would that even work? Hang the gown, spray the bleach on it, and let it drip off?) You sterilize fabrics with bleach + water + detergent + heat + agitation — with the goal not being to lyse the spores/other germs, but rather to detach all the contaminants from the fabric and suspend them in the water — which then gets flushed away.

In theory, bleach could help decrease the adhesion of the spore to a surface. A possible mechanism would be if it oxidized — and so weakened/destroyed — some spiky organic hooks that the spores were using to adhere to the fabric.

Of course, agents other than bleach — things not normally considered biocides, in fact — would likely be a lot more effective at removing spores during fabric washing, since the goal is detachment, not lysing the spore.

The obvious things (detergents themselves, and other soaps) would work, of course, to varying degrees.

But also, less-obvious things could provide benefits here. For example, if spores tended to stay adhered to fabrics because they possessed a rough proteinous exosporium that acted sort of like nano-scale velcro, then conditioners (yes, like the kind you use in hair) might get that protein coat to relax and lay flatter, in a way that disrupts the velcro-like effect.

Lubricants might also work, by "filling up" the rough valleys of the spore's surface. (Of course, you'd then need an extra wash cycle to remove the lubricants.)