I've decided to expand upon the observations on the comments.
I haven't wrapped my head yet around the idea that $A-BC$ is a 0-1 matrix. Thus I assume you or someone else has a proof for that part. Then this matrix differs from $A$ in at most r rows. But if r is less than n/k, where k is smallest such that kw is a multiple of n, then (using that $A$ is cyclic) there is a set of k rows of $A-BC$ which add up to a nontrivial multiple of the row of all ones, and thus $A$ and $A-BC$ have determinants which are not one. This handles some of the cases and reveals some of the number theory going on here.
Gerhard "Number Theory To The Rescue?" Paseman, 2019.03.05.