  This paper carries forward a series of articles describing our enterprise to
construct a gauge equivalent for the $\theta$-deformed non-commutative $p^{-2}$
model originally introduced by Gurau et al. arXiv:0802.0791. It is shown that
breaking terms of the form used by Vilar et al. arXiv:0902.2956 and ourselves
arXiv:0901.1681 to localize the BRST covariant operator $(D^2\theta^2D^2)^{-1}$
lead to difficulties concerning renormalization. The reason is that this
dimensionless operator is invariant with respect to any symmetry of the model,
and can be inserted to arbitrary power. In the present article we discuss
explicit one-loop calculations, and analyze the mechanism the mentioned
problems originate from.
