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.

I think that this works, but for some reason I have a weird feeling that I'm making some really, really stupid error!

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.