I assume hyd peroxide wouldn't work because it's actually bacteria?
Hydrogen peroxide (H202) works like a charm - I've used it several times over the years.
The method to use is specific - turn off filtration, & reduce water level if need be, to allow you to dose the H202 directly onto the cyanobacteria at close range, using a plastic syringe or test kit pipette.
Leave the filtration off for at least 1/2 an hour after dosing to allow the H202 to do its work. You'll see the oxygen bubbles adhering to the dosed cyano & then slowly lifting off, to rise to the surface.
Depending on how much cyano you have, you may need to do the dosing again over 2 days or more to get rid of it all.
The day after dosing, the cyano should have disappeared.
The primary approach for preventing it from coming back is to improve your tank cleansing routine (e.g. substrate vacuuming), and improve the water circulation within the tank, especially in the lower tank areas where the cyano generally develops & spreads.
A power bar, or a small circulation pump would be good.
Your fish & beneficial bacteria will not at all be affected so long as you limit the dosing to a reasonable quantity at a time, depending on the water volume of your tank, and of course don't spray it directly at any fish. It dissipates harmlessly very quickly.