If you are okay working with the Hitchin map defined on the moduli stack of Higgs bundles rather than the coarse moduli space, then you can find a proof of this statement in the paper [The global nilpotent variety is Lagrangian][1] by V. Ginzburg, at least in the case that the genus of the base curve is at least $2$. The precise reference is Corollary 9. 

The idea is to show that the global nilpotent cone (the most singular fiber of the Hitchin map) has the same dimension as $Bun_G$. This implies that the Hitchin map is equidimensional and that the stack of Higgs bundles $T^*Bun_G$ is lci. Since the Hitchin base is non-singular, this implies flatness by miracle flatness ([Stacks Project Lemma 00R4][2]). Now the stack of semi-stable Higgs bundles is an open substack so the restriction of the Hitchin map to this locus is also flat. 


  [1]: https://arxiv.org/pdf/alg-geom/9704005.pdf
  [2]: https://stacks.math.columbia.edu/tag/00R4