the only true non chemical way to actively kill cyano that i know of is UVC, as in a germicidal bulb(since uv sterilizer wont actually shine on it). other than that, its the usual methods of decreasing light, trying to eliminate phosphates, raising nitrates(but i dont think that will work in your case...), etc.
it wouldnt fix the underlying issue though...
removing it physically helps. if it is running on a low amount of nutrients, removing it keeps whatever is locked up in it out of the column.
im not sure how antibiotics will affect the live rock. it would worry me, but i have never tried it, so i cant say if there is any danger.