  The standard nonperturbative approaches of renormalization group for tensor
models are generally focused on a purely local potential approximation (i.e.
involving only generalized traces and product of them) and are showed to
strongly violate the modified Ward identities. This paper as a continuation of
our recent contribution [Physical Review D 101, 106015 (2020)], intended to
investigate the approximation schemes compatibles with Ward identities and
constraints between $2n$-points observables in the large $N$-limit. We consider
separately two different approximations: In the first one, we try to construct
a local potential approximation from a slight modification of the Litim
regulator, so that it remains optimal in the usual sense, and preserves the
boundary conditions in deep UV and deep IR limits. In the second one, we
introduce derivative couplings in the truncations and show that the
compatibility with Ward identities implies strong relations between
$\beta$-functions, allowing to close the infinite hierarchy of flow equations
in the non-branching sector, up to a given order in the derivative expansion.
Finally, using exact relation between correlations functions in large
$N$-limit, we show that strictly local truncations are insufficient to reach
the exact value for the critical exponent, highlighting the role played by
these strong relations between observables taking into account the behavior of
the flow; and the role played by the multi-trace operators, discussed in the
two different approximation schemes. In both cases, we compare our conclusions
to the results obtained in the literature and conclude that, at a given order,
taking into account the exact functional relations between observables like
Ward identities in a systematic way we can strongly improve the physical
relevance of the approximation for exact RG equation.
