I would think that it's effectively impossible to modify cyanobacteria to be unaffected by "its cyanophage" (it would be many different cyanophages, wouldn't it?) in the long run - you can create a strain that's immune to the currently popular cyanophages, but viruses and phages do mutate quickly and given a cyanobacteria bloom I'd expect another functioning phage to emerge soon.

Also, it seems to me that a major cyanobacteria explosion would be sufficiently fundamental to screw up the whole ecosystem around them; it might help carbon capture but at a cost/risk to other green values such as biodiversity and security of the ocean-based food chain for humans.