  This paper provides a systematic study of gauge symmetries in the dynamical
fuzzy spectral triple models for quantum gravity that have been proposed by
Barrett and collaborators. We develop both the classical and the perturbative
quantum BV formalism for these models, which in particular leads to an explicit
homological construction of the perturbative quantum correlation functions. We
show that the relevance of ghost and antifield contributions to such
correlation functions depends strongly on the background Dirac operator $D_0$
around which one perturbs, and in particular on the amount of gauge symmetry
that it breaks. This will be illustrated by studying quantum perturbations
around 1.) the gauge-invariant zero Dirac operator $D_0=0$ in a general
$(p,q)$-model, and 2.) a simple example of a non-trivial $D_0$ in the quartic
$(0,1)$-model.
