Assume $A$ is a connected (not necessarily commutative) non-semisimple Frobenius algebra that is finite dimensional over a field of characteristic 0 and given by quiver and relations. (for the commutative case all this reduced to be a local commutative Frobenius algebra that is not a field).

We should have $c(1)= \sum_{i} y_i \otimes x_i$, where $x_i, y_i$ for $i=1,...,dim A$
are defined by the condition $b (x_i y_j )= \delta_{i,j}$ (the existence of such $x_i, y_i$ is equivaleng to $A$ being a Frobenius algebra, see lemma 2.11 in the book of Lorenz on representation theory)
Now $\mu c(1)= \sum y_i x_i$ (see section 9.1.4 in the book by Lorenz) and
$b (\mu c(1))= \sum b(y_i x_i)=dim A$.
This implies (see Proposition 1.10.18 in the book on representation theory by Zimmermann) that $\mu c(1) /dim A$ is in the socle and thus squares to zero.

Thus the statement is true in the case of commutative local Frobenius algebras when the algebra is not a field extension.

In the non-local case it should be wrong by taking for example $K \times K[x]/(x^2)$ as suggested in the comments by Will Sawin.

By the way for local Frobenius algebras this window element has a purely combinatorial interpretation up to a scalar, namely it is the unique longest (in terms of product of arrows) non-zero element in the quiver algebra.