A few things I'd do:
1. Increase oxygen content of the water. Oxygen saturation is most easily accomplished by increasing water movement - particularly surface agitation that facillates gas exchange. Skimming is another way to saturate your water with O.
2. Limit your nutrient input, particularly phosphate and nitrate. Cyanobacteria (with some exceptions of nitrogen fixing species) needs a nitrogen source to conduct life process function. This means reduce your ammonia, nitrite, nitrate levels by
a. skimming out dissolved orgnanics before they break down into NH4, NO2, and NO3.
b. use R/O, DI, or distilled water to top off the evaporation and water changes. This will help remove all the aforementioned compounds, including phosphate.
c. reduce feedings. feed sparingly, targeting whatever your feeding effectively.
d. If anything is visibly decaying (coral, dead fish, macroalgae, etc.), remove it ASAP. Decaying matter frees trapped nitrogenous compounds that cyano can use.
e. Carbon will work in absorbing dissolved organics before they're broken down into available nutrients.
I'd also recommend you siphon out as much as you can of the cyano, without causing too much disturbance; do not stir up the substrate! Make sure to replace the water with fresh seawater or artificially mixed R/O water.
IME, dripping kalkwasswer works moderatley well against cyanobacteria. Not only does kalkwasser bind phosphate and precipitate it out of solution, i've also noticed raising pH and carbonate hardness retards cyano blooms. It's anecdotal, I know. But it always seems like the case for my tank.
Last point to make: wash your hands with anti-bacterial soap anytime you come in contact with cyanobacteria. Though most species are harmless, some are toxic. Toxicity depends on the species, and range from mild skin discomfort to hep
atotoxins (liver) to neurotoxins (brain). It's just good practice to wash your hands well after you're done working on your tank.