In order to derive the inhomogeneous Maxwell equations, you do not need to use the antisymmetry of the field strength tensor. They follow from the equations of motion for $A_\mu$. If you start with
$$ \mathscr{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-e J^\mu A_\mu\;,$$
the Euler-Lagrange equations give you
$$ \partial_\mu F^{\mu\nu}=e J^\nu$$
up to a total derivative term that can be gauged or argued away. If you relabel $\mu\leftrightarrow\nu$, you get
$$ \partial_\nu F^{\nu\mu}=e J^\mu\;.$$
It might very well be that our conventions differ because I do not work with these $\mu_0$ oddities. That is, all the above assumes natural units, in which $c=1$. In any case, you do not get a sign when renaming the summation indices.