I want to compute the stress-energy tensor for the following Lagrangian: $$\mathcal{L} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} - \frac{1}{2\xi} (\nabla_\mu A^\mu)^2$$
but I'm struggling with the gauge-fixing term. When taking variations with respect to the metric, it seems that it will involve many terms, some of then not clearly covariant. I am only interested in its value in Minkowski, but that is also hard to find.
By stress-energy tensor I mean the Hilbert one, defined as:
$$T_{\mu \nu} = \frac{-2}{\sqrt{-g}} \frac{\delta S}{\delta g^{\mu \nu}}$$
My attempt
I have attempted to include an auxiliary field, such that
$$\mathcal{L}' = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + B \nabla_\mu A^\mu + \frac{\xi}{2} B^2$$
which becomes the original one after finding and imposing the equations of motion for $B$. But is it true that the two stress energy tensors are going to be the same? What I find is that
$$T_{\mu \nu} = \eta_{\mu \nu} \mathcal{L} + F_{\alpha \mu} F_{\beta \nu} g^{\alpha \beta} + \frac{4}{\xi} \partial_\alpha A^\alpha \partial_{(\mu} A_{\nu)} + \frac{2}{\xi} \partial_{(\mu} \left(\partial_\alpha A^\alpha \right) A_{\nu)} - \frac{1}{\xi} \eta_{\mu \nu} \partial_\rho(A^\rho \partial_\alpha A^\alpha)$$
in Minkowski, and after getting rid of the auxiliary field. Can this be correct? What if I integrate by parts the first term involving the auxiliary field, so that I get:
$$\mathcal{L}' = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} - \nabla_\mu B A^\mu + \frac{\xi}{2} B^2 = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} - \partial_\mu B A^\mu + \frac{\xi}{2} B^2 $$
given that $B$ is a scalar field. This seems to be giving me a different stress-energy tensor as well.