  This paper aims at investigating the nonperturbative functional
renormalization group for tensorial group field theories with nontrivial
kinetic action and closure constraint. We consider the quartic melonic
just-renormalizable $[U(1)]^6$ model and show that due to this closure
constraint the first order Ward-Takahashi identity takes the trivial form as
for the models with propagators proportional to identity. We then construct the
new version of the effective vertex expansion applicable to this class of
models, which in particular allows to close the hierarchical structure of the
flow equations in the melonic sector. As a consequence, there are no additional
constraints on the flow equations, and then we can focus on the existence of
the physical Wilson-Fisher fixed-points in the symmetric phase.
