Let $k$ be a field, $\bar{R} \to R$ a local homomorphism of artinian local rings with the residue fields $k$, $I$ its kernel, $A/R$ an abelian scheme, and $\mathscr{T}$ its tangent sheaf. Let $A_0 = A \times_R k$. Assume that $\mathfrak{m}_\bar{R} I = 0$. Then $H^2(A, \mathscr{T}_{A/R} \otimes_R I) \cong H^2(A_0, \mathscr{T}_{A_0/k}) \otimes_k I$?

This is a part of the proof of (2.2.4.1) of Kai-Wen Lan's "Arithmetic Compactifications of PEL-Type Shimura Varieties". To show it, I need the following proposition:

Let $S$ be a scheme, $f : A \to S$ an abelian scheme of relative dimension $g$. Then the sheaf $R^pf_* \Omega^q$ is locally free. And this formation commutes with any base change.

Are there "elementary" proof of this?

I know this is (2.5.2) of Berthelot, Breen, Messing's Théorie de Dieudonné Cristalline. But its proof is too hard for me, since it heavily relies on the theory which I don't know.

And I know that this post shows it elementary. But it uses the formally smoothness and the pro-representability of the deformation of "abelian schemes + polarization", which is what I want to show using this highlighted statement. So it is a circular reasoning for me.