Seedlings are looking good,
I have never used bleach but I've read many people do. I use a 5 gallon paint strainer bag in all my 5 gallon buckets. At the end of the growing season I rinse all out with fresh water and let it dry up on the ground in the strainer bag. When dry I pour the perlite out of the bags into an extra large trash can and leave it out for the winter then do everything all over again for the following spring.
Since you're down in a warm climate all year that wont work so you are probably better off using a little bit of bleach. I would check the perlite at the bottom of the container first to see if it needs cleaning.