I agree that there are phototrophic organisms that don't produce oxygen. I apparently didn't use the term "photosynthetic" quite properly. Sigh!
I agree that growing them would be difficult, and I'm not sure why this would be desirable. Are you sure that deoxygenating the water would be enough to prevent the growth of oxygen-producing photosynthetic organisms? I wasn't clear on that. I am not an expert in biology, by any means.