The Brauer groups of moduli of genus three curves and plane curves

We compute the -primary torsion of the Brauer group of the moduli stack of smooth curves of genus three over any field of characteristic different from two. We achieve this result by computing the cohomological invariants of this stack. Along the way, we also compute the cohomological invariants and the Brauer group of the moduli stacks of smooth plane curves of degree d over any algebraically closed field of characteristic different from two and coprime to d.