  In this paper, we study rewriting modulo a set of algebraic axioms in
categories enriched in linear categories, called linear~$(2,2)$-categories. We
introduce the structure of linear~$(3,2)$-polygraph modulo as a presentation of
a linear~$(2,2)$-category by a rewriting system modulo algebraic axioms. We
introduce a symbolic computation method in order to compute linear bases for
the vector spaces of $2$-cells of these categories. In particular, we study the
case of pivotal $2$-categories using the isotopy relations given by
biadjunctions on $1$-cells and cyclicity conditions on $2$-cells as axioms for
which we rewrite modulo. By this constructive method, we recover the bases of
normally ordered dotted oriented Brauer diagrams in te affine oriented Brauer
linear~$(2,2)$-category.
