  We present an analysis of the perturbative realization of the $TTJJ$
correlator, with two stress energy tensors and two conserved currents, using
free field theory realizations, integrating out conformal sectors in the
quantum corrections. This allows defining, around flat space, an exact
perturbative expansion of the complete anomaly effective action - up to 4-point
functions - whose predictions can be compared against those of the anomaly
induced action. The latter is a variational solution of the conformal anomaly
constraint at $d=4$ in the form of a nonlocal Wess-Zumino action. The
renormalization procedure and the degeneracies of the tensor structures of this
correlator are discussed, valid for a generic conformal field theory, deriving
its anomalous trace Ward identities (WIs). In this application, we also
illustrate a general procedure that identifies the minimal number of tensorial
structures and corresponding form factors for the $TTJJ$ and any $4$-point
function. The result of the direct computation is compared against the
expression of the same 4-point function derived from the nonlocal anomaly
induced action. We show that the prediction for the anomalous part of the
$TTJJ$ derived from such action, evaluated in two different conformal
decompositions, the Riegert and Fradkin-Vilkovisky (FV) choices, differ from
the anomaly part identified in the perturbative $TTJJ$, in the flat spacetime
limit. The anomaly part of the correlator computed with the Riegert choice is
affected by double poles, while the one computed with the FV choice does not
satisfy the conservation WIs. We present the correct form of the expansion of
the anomaly induced action at the second order in the metric perturbations
around flat space that reproduces the perturbative result.
