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. 

**Edit:** I think the result also holds for the coarse moduli space of semi-stable Higgs bundles but it seems to be a bit subtle. I think equidimensionality of the coarse Hitchin map follows from that on the stack. Then by miracle flatness the map is flat if and only if the coarse space is Cohen-Macaulay. Locally the coarse space of semi-stable Higgs bundles looks like a quotient $V//H$ where $H$ is reductive and $V$ is affine chart for the stack (in the smooth topology). Thus the question becomes when is $V//H$ Cohen-Macaulay? 

For $V$ non-singular this is the Hochster-Roberts theorem. However, it can fail in general even when $V$ itself is Cohen-Macaulay (in fact even when $V$ is a complete intersection). See for example the last paragraph of example $I$ [here][3]. In this case we are saved by the fact that moduli space of semi-stable Higgs bundles has symplectic singularities which are in particular Cohen-Macaulay. See for example [this paper.][4] 

It seems to me then that being in the symplectic setting is used not just for the dimension bounds but also to ensure that the moduli space is Cohen-Macaulay so I'm not sure what to expect for Higgs bundles valued in an arbitrary line bundle $L$. 

  [1]: https://arxiv.org/pdf/alg-geom/9704005.pdf
  [2]: https://stacks.math.columbia.edu/tag/00R4
  [3]: https://projecteuclid.org/euclid.bams/1183541144
  [4]: https://arxiv.org/pdf/1701.07468.pdf