Although I agree bleach is one option to purge slime, I don't agree that it can be ran with organics for any real amount of time. That would depend on just how organic the nutes are I suppose. Without microbes the organics will not get broken down, and therefore can not be absorbed by roots. Roots only absorb nutrients in chemical form. So yes, organics eventually become chemicals.
If your only option is to continue running organics, then your sterilization followed by re-inoculation plan seems to be your best bet. Simply adding extra molasses to your tea will not boost it's power though, and in fact could result in left over food that the slime will eat. To fight slime, you need diversity, which comes best from great white + ancient forest compost. I think if you overcome the slime problem and are diligent with the tea, you can be successful; I just personally find organics to be an unnecessary factor.