First choice would be a uv. (for longterm solution)
Second choice would be chemicals. ((algae fix by Aquarium Pharmaceuticals "kills green water" )short term until it returns)
Definately stop the large water changes 50% plus as these will only provide the "illusion" of a fix.
Definately do not clean your filters as your assumption is totally correct.
I have used and sold this product many times without any detrimental affects.

Second choice would be chemicals. ((algae fix by Aquarium Pharmaceuticals "kills green water" )short term until it returns)
Definately stop the large water changes 50% plus as these will only provide the "illusion" of a fix.
Definately do not clean your filters as your assumption is totally correct.
I have used and sold this product many times without any detrimental affects.

I'd say the problem is solved by now. Except for people not being able to read basic English. That problem clearly is not solved...