Is there a transitive permutation group satisfying these properties?

Question

Let $G \subset S_n$ be a transitive permutation group and let $H=G_1:=\{ g \in G \ \vert \ g(1)=1 \}$.
Let $(K_i)_{i \in I}$ be the sequence of minimal overgroups of $H$ in $G$.
Note that if $G$ is primitive, then the sequence above is reduced to $G$ alone.

Let the following properties:

1. There are two distinct irreducible representations $V$ and $W$ of $G$ such that $dim(V^H)$ and $dim(W^H) \ge 2$.
2. $\sum_{i \in I} [G : K_i] > [G:H]$ $ (= n)$.
3. The lattice of intermediate subgroups $\mathcal{L}(H \subset G)$ is distributive.

Question: Is there a transitive group satisfying ($1$), ($2$) and ($3$)?
If yes, what are the first examples?
Or anyway, what's your better lower bound for $n$?

Remark: Thanks to a GAP computation, the answer is no for $n \le 17$.
So my better lower bound is $n \ge 18$.