This is one of the issues with Chaeto as an export algae, IME. Under some tank conditions hair algae growing on it as an epiphyte may overwhelm it.
I have no method to kill hair algae without effecting the Chaeto. They are in competition for nutrients and the hair algae is winning. You need to shift the nutrient dynamics so that hair algae is no longer dominant.
If its a brightly lit sump I would try floating Sargassum or Caulerpa in it along with the Chaeto. Given enough light, Sargassum and Caulerpa have very high nutient uptake capabilities and both reject epiphytes better than Chaeto IME. Its possible that the other macroalgae will soak up enough nutrients that the hair algae collapses, allowing the Chaeto to survive.