It is an exercise with descent theory and the snake lemma for fppf abelian group sheaves to deduce the result for $G[n]$ from the cases of $T[n]$ and $A[n]$.
In more detail, by the snake lemma $G[n]$ is an extension of $A[n]$ by $T[n]$ in the sense of such abelian sheaves. Since $A[n]$ and $T[n]$ are each finite fppf over $S$, the same then holds for $G[n]$. Indeed, rather generally, if
$$1 \rightarrow G' \rightarrow G \rightarrow G'' \rightarrow 1$$
is a complex of $S$-group schemes with $G'$ affine fppf over $S$ and the diagram is short exact for the fppf topology (so $G'$ is the scheme-theoretic kernel of $G \rightarrow G''$) then the functor of points of $G$ as a $G''$-scheme is a $G'$-torsor for the fppf topology on $G''$, so the $G''$-scheme $G \rightarrow G''$ becomes isomorphic fppf-locally on $G''$ to $G'$ (over the base) as a scheme. Hence, by fppf descent for properties of morphisms, $G \rightarrow G''$ inherits many "nice" properties that may be satisfied by $G' \rightarrow S$, such as: proper, flat, smooth, etale, finite, etc. In particular, $G$ is fppf over $G''$ and if $G'$ is finite over $S$ then so is $G \rightarrow G''$ (and hence so is $G$ if $G''$ is
also finite over $S$).
See Oort's LNM book on commutative group schemes for generalizations with the fpqc topology (around section 18, IIRC).