As Bass and Nualart mention (Theorem 2.2.1 in "MC and related topics"), they are talking about the matrix derivative DX, not the Malliavin derivative $D_{h}X$. That's what Bass means by "where we use the notation of (I.10.4).".

To the get Malliavin Lp-smootheness, he uses the approximation result (2.2) Lemma by the Lp smooth solutions built in the proof for both the X and DX sdes.

As Bass and Nualart mention (Theorem 2.2.1 in "MC and related topics"), they are talking about the matrix derivative DX, not the Malliavin derivative $D_{h}X$. That's what Bass means by "where we use the notation of (I.10.4).".

To the get Malliavin Lp-smootheness, he uses the approximation result (2.2) Lemma by the Lp smooth solutions built in the proof for both the X and DX sdes.

He doesn't explicitly work with Frechet derivatives but mostly leaves it as exercise at the step "it is easy to see that... is Frechet differentiable". Indeed, since we know that $Y$ is Lp-smooth, we can just use the product rule version for Malliavin.