Let $X \subset \mathbb{R}^n$, $Y \subset \mathbb{R}^m$ and $\Gamma: X \rightarrow Y$ a correspondence, are there sufficient conditions so that there exists a function $h:X \rightarrow Y$ such that $h$ is differentiable and $h(x) \in \Gamma(x)$?
In the particular context that I am interested $\Gamma(x)$ would be the set of solutions of
$$\max_{y \in S(x)} f(y)$$
Where the set that $f$ is being maximized depends of $x.$
Aubin has a book called Set-Valued Analysis that you could look at. It has many results about the properties of differential inclusions. Here's a link to one of his lectures: https://www.math.lsu.edu/system/files/control-seminar-Sept26.pdf
Clarke works a lot on optimal control when the change in the state belongs to a set. Maybe you can repose your problem in that framework?
Rockafellar has a book on Implicit Function Theorems that has a bunch of content about inclusions, as solutions to an equilibrium system.
I would ask myself how much I really needed differentiability: it would be might be much easier to, for example, use Michael's selection theorem to get a continuous selection, then use other details of the problem to prove something like Lipschitz continuity, then use that to prove almost everywhere differentiability or a related property. I've found Aubin's stuff to be somewhat impractical for applied problems, but maybe you'll be luckier.