The LHS might not make sense if g is very irregular. 

Consider, pth-Holder function $g(x)=x^{p}$ for $0<p<1$ and $\frac{1}{p}$ odd integer and Brownian motion $X_{t}=B_{t}$, then the process $g(X_{t})= (B_{t})^{p}$ for $p<\frac{1}{3}$ is not even in the usual regime for rough integrals (https://mathoverflow.net/questions/366379/uniqueness-of-solutions-of-young-differential-equations/435555#435555). So we would also need to formulate the LHS in some way.

Generally, we also need $C^2$/Convex in order to apply change of variables see here https://almostsuremath.com/2020/10/12/the-ito-tanaka-meyer-formula/


In even weaker settings of rough paths, there is still an Ito formula (In Proposition 6.9 ["Rough Path Theory"][1] by Andrew L. Allan).

[![enter image description here][2]][2]

Even for just Brownian motion for the Ito-formulation, in most cases of $g$ there will be a drift-term (if there was a formula with no-drift term, by taking expectation or using Ito-isometry we would likely get contradictions to the Ito formula). 

But if you interpet the RHS as the Stratonovich-formulation there is the regular chain rule

$$  g'(W_t) \circ \mathrm{d} W_t = dg(W_t),$$

and so one can use the above Ito-formulas to get something close to the OP statement (see [here][3] for semimartingales and Stratonovich). 


  [1]: https://people.math.ethz.ch/~aallan/RP_lecture_notes_Allan.pdf
  [2]: https://i.sstatic.net/4THvO.png
  [3]: https://math.mit.edu/~dws/ito/ito8.pdf