Gauge-invariance of the Maxwell Lagrangian in Srednicki's book

When it comes to the demonstration of the gauge-invariance of the Lagrangian of the Maxwell-theory Srednicki’s book proceeds as follows:

$${cal L} = -frac{1}{4}F^{munu}F_{munu} + J^mu A_{mu} tag{54.21}$$

I am interested in the second term $J^mu A_mu$. Under a gauge transformation the 4-vector potential transforms as $A’_mu =A_{mu} – partial_mu Gamma$ In order to show gauge invariance one has to show that

$$J^mu ( A’_{mu} – A_{mu}) = -J^mu partial_mu Gamma = (partial_mu J^mu)Gamma -partial_mu(J^mu Gamma)$$

can be "neglected".
The argumentation of Srednicki is that the second term can be neglected since upon considering the action the corresponding integral over $d^4x$ vanishes and the first term disappears because of current conversation. The latter, however, would mean that the Lagrangian is only gauge-invariant for the $A_mu$-field configuration that fulfills the Maxwell-equations, otherwise the current would be not conserved. Is the Maxwell-Lagrangian (54.21), in particular the coupling term, gauge-invariant for all field configuration or only for the special one that fulfills the field-equations?

In particular in view of the comments given to the post
Is the Dirac Lagrangian locally gauge invariant without gauge field $A$? I am confused.

EDIT
Actually, I don’t know if a background current would automatically fulfill the equation $partial_mu J^mu=0$, but at least if the Maxwell-equations are fulfilled, i.e. $partial_nu F^{munu} = J^mu$ by taking a second derivative we get: $0=partial_mupartial_nu F^{munu}= partial_mu J^{mu}$. So from this perspective the current conservation seems to a result of the Maxwell-equations. And this is what Srednicki mentions the page that just precedes that of expression (54.21). That suggests strongly that the current conservation only is valid if the Maxwell-equations are fulfilled.