Cyanobacteria is a gram-negative bacteria. Since it has a thin inner cell wall containing peptidoglycan (I think that's how it is spelled), the main effect of "Red Slime Remover" is probably to inhibit the synthesis of this layer of the cell.
Since bacterial cells have high inner osmotic pressure, as the cell wall gets thinner and thinner, it eventually bursts, spewing the contents of the cell into the tank water. This has two consequences that are the reasons I would never put the stuff into my tank:
#1 The reaction spews the contents of the cyanobacteria cell into the water. If you have a particularily high amount of cyano in the tank, this can foul the water, even causing drastic fluctuations in ph, or even making high nutrient concentrations even worse!
#2 The ammino acids targeted by the antibiotic in "Red Slime Remover" while unusual, are present in all bacteria that contain peptidoglycan... meaning you are wiping out almost all of your gram-negative bacteria in your tank.