Your tests aren't lying..that's not how it works!
Cyano is a maestro at using "unusable" forms of nutrients.
Better algae are not so capable.
Better algae need simple nutrients like nitrates to be available in the water.
So every water change you did would knock down the already undetectable levels of nutrients in the water even lower, making conditions even more favorable to cyano growth.
Slowing your water change schedule would temporarily let other things grow.
Thank the ULNS and related trek to dino-land for the unusual experience.
It might be a little complicated to do but if you can save your coraline scrapings and sprinkle them over the cyanobacteria areas it will help seed the area. You will have to maintain positive nutrient levels to sustain the bloom you'll want though.