Brauer groups of moduli of hyperelliptic curves via cohomological invariants

We compute the Brauer group of the moduli stack of hyperelliptic curves $\mathcal{H}_g$ over any field of characteristic zero. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.
Comment: 35 pages, comments are very welcome! New version: we extended our choice of coefficients to general cycle modules, using Gille and Hirsch's results (arXiv:1908.08146). Our methods moreover yield a more precise descriptions of the groups appearing in Gille and Hirsch's splitting formula