I would suggest that the stickiness is not in the bleach, but in the goo formed when bleach attacks mold. Also, bleach kills mold on contact but has no lasting effect. So best that you spray your air conditioner weekly or even more often, and then you should reduce or eliminate clogging problems. That’s my theory.

Diluted household bleach put into one of those hand pumped sprayers works very well. It kills the mould for about 3 months. Only snag is that if the fins are very close together, in time, the bleach which is sticky tends to clog the airflow through the fins. You then need to try and remove the residue with steam or maybe a hot water spray.
You can also use ordinary mould remover bought at your local supermarket. It behaves in the same way as bleach

Sounds like you’ve found some effective ways to remove the mold, but haven’t yet found the root cause. Mold needs moisture and something to grow in, usually dust in the case of air conditioners.

The moisture comes from condensation so there isn’t much you can do about that other than make sure the fins are vertical so the water drips off and that the water drains straight out of the unit.

Have you checked the air filter to make sure it is preventing dust from entering the unit?

Maybe there is something about the particular model of air conditioner that you have, have you googled mold problem + your air conditioner make and model?

Hope that helps