If you are worried about dead cyano fouling up your water, you may opt to be on the safe side and remove some cyano before treatment as Jim said, and do a water change as soon as cyano appeared to have died off. They should disappear in a day or two. As I mentioned earlier, Red Slime Remover treatment was quite safe followed by a weekly water change.
By the way, Chemi Pure will not work for the red bubble algae. Red bubble algae is tough to eliminate. They can persist in pristine water, too. WEEC was full of them not too long ago and EvilMel was also struggling with them a little while ago. Luckily my foxface rabbitfish ate all of mine, but a rabbitfish is too big for your nano. Can you take the rock out of the nano? You can remove them with tweezers and scrub the remnants off with a tooth brush really well. Be sure to rinse the rock with a plenty of water before putting it back in your tank.
Tomoko