I have read through Jackson and also Classical field theory (Florian Scheck) on this topic and neither of them address my question. In both of them we are told that the canonical stress energy tensor is given by
$T^{\alpha \beta}=\frac{\partial \mathscr{L}}{\partial\left(\partial_{\alpha} A^{\lambda}\right)} \partial^{\beta} A^{\lambda}-g^{\alpha \beta} \mathscr{L}$
We go on and derive it for a source free environment so the Lagrangian density contains no current term. From this we also derive the symmetric stress energy tensor. But, when we move to the situation where there is a source current, the symmetric stress energy tensor is exactly the same. What confuses me about this is that it seems that they do not include the current from $\mathscr{L}=-\frac{1}{16 \pi} F_{\alpha \beta} F^{\alpha \beta}-\frac{1}{c} J_{\alpha} A^{\alpha}$.
What I'm expecting is
$\Theta^{\alpha \beta}=\frac{1}{4 \pi}\left(g^{\alpha \mu} F_{\mu \lambda} F^{\lambda \beta}+\frac{1}{4} g^{\alpha \beta} F_{\mu \lambda} F^{\mu \lambda}\right)+\frac{g^{\alpha\beta}}{c}J_{\mu}A^{\mu}$
but what they're saying is
$\Theta^{\alpha \beta}=\frac{1}{4 \pi}\left(g^{\alpha \mu} F_{\mu \lambda} F^{\lambda \beta}+\frac{1}{4} g^{\alpha \beta} F_{\mu \lambda} F^{\mu \lambda}\right)$
What I've done so far is try to show that extra term goes to zero once differentiated so it doesn't effect the conservation laws but haven't been unsuccessful.