  We continue the program started in \cite{M1} to understand the combinatorial
commutative algebra of the projective coordinate rings of the moduli stack
$\mathcal{M}_{C, \vec{p}}(SL_2(\C))$ of quasi-parabolic $SL_2(\C)$ principal
bundles on a generic marked projective curve. We find general bounds on the
degrees of polynomials needed to present these algebras by studying their toric
degenerations. In particular, we show that the square of any effective line
bundle on this moduli stack yields a Koszul projective coordinate ring. This
leads us to formalize the properties of the polytopes used in proving our
results by constructing a category of polytopes with term-orders. We show that
many of results on the projective coordinate rings of $\mathcal{M}_{C,
\vec{p}}(SL_2(\C))$ follow from closure properties of this category with
respect to fiber products.
