I'm not suggesting that this will damage anything, I just don't see how this will help overall with a green pool. Adding chlorine to the skimmer is not going to clean the filter media. Algae, whether dead or alive, is still going to clog it. The root cause of algae in the filter is the algae in the pool.
I'm not trying to argue with you, but "adding a small amount just because" is not how we advocate pool care here at TFP. Besides, the chlorine will pass through so quickly it will have next to no effect on the algae in the filter. Killing the algae in the main body of water should be the focus, and that is done by maintaining a specific FC level over time through a SLAM, not from a quick burst of chlorine through the filter which seems familiar to the "shock & pray" methods used by pool stores.
Our methods are based on facts, and have been proven in thousands of pools throughout this forum. It seems that you "think" that by killing a small % of algae in the filter will prevent the filter from being clogged again from the majority of the algae that is still in the water.
Dom