Try using an empty spray bottle with the bleach solution, don't saturate it and don't touch it while it is wet. It may take a couple applications, wait until the first is dry to see if it needs another.

Most of the "popcorn" or acoustic texture is sprayed on and left as-is. If it is getting damp enough to grow mildew you may want to consider painting it with a quality bathroom paint.

Another option (and I think better one) is to remove the texture all together. It's a messy job, but you only do it once. Then smooth and paint the ceiling with good paint and be done with the problem forever.
To remove the texture, wet it with a mist sprayer and scrape it off with a putty knife. Be sure to cover everything with plastic first.

You may also consider having a company which specializes in popcorn ceilings check it out.There are several options available including recoating.If you have other popcorn ceilings in the house you may find a good deal by having them all spruced up at once.