I love bleach. I buy it by the case. I always use it to wash puppy pads, even if they say not to. I do use the "extra rinse" cycle for puppies though. I wish i could use bleach on my darks, too. Love it.
You might try Baking Soda and do an extra rinse. I use bleach and sometimes soda. They have some Bleach that are scented and rinse once or twice to make sure they are clean.
I throw Simple Solution into my washer. I don't have pups but I use it whenever I wash the dog bedding and towels and the like. I put maybe 1/4 cup into the detergent container with the detergent and about the same into the softener container of the tray.
I use bleach too, but I also use liquid fabric softener. I'm addicted to the smell of either Downy or Gain Apple Mango Tango. My fleece holds up just fine - I've used bleach for years, plus it kills germs.