  Brundan, Kleshchev and Wang equip the Specht modules $S_{\lambda}$ over the
cyclotomic Khovanov--Lauda--Rouquier algebra $\mathscr{H}_n^{\Lambda}$ with a
homogeneous $\mathbb{Z}$-graded basis. In this paper we begin the study of
graded Specht modules labelled by hook bipartitions $((n-m),(1^m))$ in level
$2$ of $\mathscr{H}_n^{\Lambda}$, which are precisely the Hecke algebras of
type B, with quantum characteristic at least three. We give an explicit
description of the action of the Khovanov--Lauda--Rouquier algebra generators
$\psi_1,\dots,\psi_{n-1}$ on the basis elements of $S_{((n-m),(1^m))}$.
Introducing certain Specht module homomorphisms, we construct irreducible
submodules of these Specht modules, and thereby completely determining the
composition series of Specht modules labelled by hook bipartitions for
$e\geqslant{3}$.
