Bleach will indeed hasten the deterioration of the material. People hand-washing and rinsing the insides of a PUP cannot remove all of the bleach and therein lies the problem.
I have had good luck with Iosso Mold and Mildew Stain Remover. It is designed for woven materials (primarily sails for sailboats) and is non-toxic, doesn't smell (important when you are inside a PUP) and is biodegradable. I googled a supplier and have been quite pleased with it.