Let $R=k[x,y,z]/\left((xy-1)z\right)$, for $k$ some field. Let $A:R\to R$ be multiplication by $z$. Let $B:R\to R$ be multiplication by $xz$. Then $A$ and $B$ have the same image, since $z=xyz$, and therefore isomorphic cokernels. They also have the same kernel, namely the set of polynomials $p(x,y)$ that are multiples of $xy-1$. The only units in $R$ are scalars, so there are no isomorphisms that complete a commutative square.