The Ward identity, the Lorentz invariance Outline - heuristic derivation of the Ward identity from the requirement of the Lorentz invariance
Suppose we have the free quantized gauge theory (with quanta called photons) with the Hilbert space restricted to transversal polarizations only. Next, suppose interactions with the matter, and assume the matrix element $M$ with (at least) one external photon line having the polarization vector $\epsilon^{\mu}_{\perp}(p)$. It has the form
$$
\tag 1 M = \epsilon^{\mu}_{\perp}(p)M_{\mu}
$$
Assume $M_{\mu} \sim p_{\mu}$. Next, perform the massless Lorentz orbit little group transformation $\Lambda$ defined by
$$
\tag 2 (\Lambda p) = p, \quad (\Lambda \epsilon_{\perp_{1}}) = \epsilon_{\perp_{1}} +cp, \quad (\Lambda \epsilon_{\perp_{2}}) = \epsilon_{\perp_{2}} 
$$
and apply it to transformed squared averaged matrix element 
$$
|\bar{M}|^{2} \equiv \sum_{\perp_{1,2}}|M|^{2}
$$ 
Since it must be invariant in the Lorentz-invariant gauge theory, we obtain
$$
|\bar{M}|^{2} = |\bar{M}^{2}| + c(p^{\mu}\epsilon_{\perp_{1}}^{*\nu} + p^{\nu}\epsilon_{\perp_{1}}^{\mu})M_{\mu}M^{*}_{\nu},
$$
from which follows the requirement
$$
\tag 3 p^{\mu}M_{\mu} = p^{\nu}M_{\nu}^{*} = 0,
$$
i.e., the Ward identity.
My question
1) What is the physical sense of the transformation $(2)$? Precisely, may I state that it generates the unphysical "longitudinal" state from the physical transverse one? 
2) Next, does from the derivation above follow that the unitarity (a-la optical theorem) is broken as long as $(3)$ isn't valid? Precisely, being inserted as the sub-diagram the amplitude $M_{\mu}$ will distinguish physical and unphysical polarizations, and that's enough for proving the non-unitarity (see this related question)? For example, one may calculate the optical theorem (with $A$ denoting the state containing the photon) 
$$
2\text{Im}M_{A \to A} = (2\pi)^{4}\sum_{n}|M_{A \to n}|^{2},
$$
where for the given order of perturbation theory $M_{A \to A}$ doesn't respect the Ward identity while $M_{A \to n}$ respects it. Then one may perform the transformation $(2)$ and deduce that there is indeed the violation of the unitarity.
3) Finally, does this relate directly the Lorentz invariance to the unitarity of the gauge theory?
 A: 1) see Weinberg's book.
2) You (and some introductory texts) have the logic inverted. Your "derivation" of $(3)$ is not a derivation of the Ward identity; it only serves to explain why the Ward identity is important. A quick proof of the Ward identity can be found in Srednicki's book.
3) If you use the Coulomb gauge then the Ward identity guarantees that the theory is covariant. If it is not satisfied, then covariance is broken but the theory is still unitary: there are no negative norm states.
If you use the $R_\xi$ gauge then the Ward identity guarantees that the theory is unitary. If it is not satisfied, then unitarity is broken but the theory is still covariant: the polarisation vectors are all true vectors, but the longitudinal ones are not decoupled. 
This means that the Ward identity is related to both covariance and unitarity, but not at the same time. If the Ward identity becomes anomalous you either lose covariance or unitarity, but not both. And you must decide which one you want to give up (the choice is irrelevant anyway because if one of these properties isn't satisfied, then the whole theory becomes meaningless).
