I have a question regarding the physical significance of the canonical energy momentum tensor $T_\nu ^\mu$ in the context of classical field theory. It is defined as
$T_\nu ^\mu = \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \Phi^I)} \partial_\nu \Phi^I - \delta_\nu ^\mu \mathcal{L} $, where $\Phi^I$ is the set of all relevant scalar, vector, or tensor fields in the Lagrangian, and $I$ is the corresponding indices. For example, if we consider the Lagrangian for the free gauge field in electrodynamics $\mathcal{L}_A = - \frac{1}{4} F_{\mu \nu} F^{\mu \nu}$, where $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$, we have that $\Phi^I = A_\mu$. Accordingly, the (covariant) canonical energy momentum tensor is
$T_{\mu \nu} = \frac{1}{4}\eta_{\mu \nu} F_{\alpha \beta} F^{\alpha \beta} - F_{\mu \alpha} \partial _ \nu A^ \alpha$.
It satisfies that $\partial _\mu T_\nu ^\mu = 0$ provided that the system, which it describes, is independent of translations in spacetime. This is a consequence of Noethers theorem.
My problem: On the one hand, it is used to define the canonical 4-momentum by
$P_\nu = \int T_{\nu}^0 d^3 x $,
i.e. it is used to define some physical observables of the system. Also, the canonical Hamiltonian $\mathcal{H}$ density is defined by $\mathcal{H}=T_{0}^0$
On the other hand, $T_\nu ^\mu$ is not in general gauge invariant - but physical observables must be gauge invariant. If the gauge field is time independent, i.e. $\partial_0 A^\alpha = 0$ for each $\alpha$, then $T_0 ^0$ (and thus also the energy) can be gauge independent, as is the case for $T_0 ^0$ for free electrodynamics defined above.
However, in general for time-dependent gauge field $T_\nu ^\mu$ is not gauge-invariant, which must imply that the canonical Hamiltonian density is not in general gauge independent and thus not observable.
My question: Is it possible to show that the 4-momentum is always gauge invariant and thus observable despite the fact that $T_\nu ^\mu$ is not?
If this is not possible, what even is the significance (and relevance) of $T_\nu ^\mu$ and the canonical 4-momentum?