The only thing you should put in your car wash bucket is…water! No soap, bleach, shampoo, conditioner, etc. After washing your car you can put a cap full of car wash in the bucket to wash your tires and wheels. The reason you wax your car is to keep dirt (and mildew) off the paint. If you wash with soap (or bleach) you are stripping off the wax. Twice a year, I use a clay bar then wax it.
Are you properly drying your car after you wash it? Mildew needs moisture to grow – make sure it’s dried soon after it’s washed. You said you are not cleaning it often enough – do it every week and see if the mildew goes away.
Twotone