Let $\Bbb{F}$ be an arbitrary field of characteristic $p\geqslant 0$. Let $e\in\{2,3,\ldots\}$ be the quantum characteristic, and identify $I\coloneq \Bbb{Z}/e\Bbb{Z}$ with the set $\{0,1,\ldots,e-1\}$. Let ${\kappa=(\kappa_{1},\kappa_{2})\in I^{2}}$ be an $e$-bicharge, and let $\Lambda=\Lambda_{\kappa}\coloneq \Lambda_{\kappa_{1}}+\Lambda_{\kappa_{2}}$ be its associated dominant weight of level~2. If $p=e$ or $p$ does not divide $e$ then the integral Hecke algebra $\Cal{H}_{n}(\zeta, Q_{1},Q_{2})$ of type $B$, with parameter $\zeta\in\Bbb{F}$ an $e$-th root of unity, $Q_{1}=\zeta^{\kappa_{1}}$, and $Q_{2}=\zeta^{\kappa_{2}}$, is isomorphic to the cyclotomic KLR algebra $\Cal{R}_{n}^{\Lambda}$ via (a special case of) Brundan and Kleshchev's {\it Graded Isomorphism Theorem}. It is known that the Specht modules for these algebras are indexed by bipartitions of $n$, and that these Specht modules are all indecomposable unless $e=2$ or $\kappa=(i,i)$ for some $i$, which, up to isomorphism, can be assumed to be 0 in this case. For each regular bipartition $\lambda$ of $n$, the Specht module $\roman{S}_{\lambda}$ has a simple head $\roman{D}_{\lambda}$, which is self-dual as a graded module. It is known that the set ${ \{\roman{D}_{\lambda}\mid\lambda\text{ is a regular partition of $n$}\} }$ is a complete set of graded simple $\Cal{R}_{n}^{\Lambda}$-modules up to isomorphism and grading shift. \par In [Part I, Pacific J. Math. {\bf 304} (2020), no.~2, 655--711; MR4062785], L. Speyer and L. Sutton found a large family of decomposable Specht modules indexed by bihooks, and conjectured that if $e\neq 2$ and $p\neq 2$, these are all of the decomposable Specht modules indexed by bihooks. In the paper under review, which is a sequel to [op. cit.], and is inspired by a recent semisimplicity result of C. Bessenrodt, C.~D. Bowman and the third author of the paper under review [Trans. Amer. Math. Soc. Ser. B {\bf 8} (2021), 1024--1055; MR4350547], the authors study the structure of these decomposable Specht modules, and determine when they are semisimple. The authors give the results in the case of bipartitions of the form $((ke),(je))$, from which the corresponding results for the more general bihooks are obtained by applying graded $i$-induction functors. Specifically, the authors show that for $k\geqslant j\geqslant 1$, $\roman{S}_{((ke),(je))}$ is semisimple if and only if one of the following holds: \roster \item"$\diamond$" $p\neq 2$ and $p$ does not divide any of the integers $k+j,k+j-1,\ldots,k-j+2$; \item"$\diamond$" $p=2$, $j=1$, and $k$ is even; \item"$\diamond$" $p=2$, $j=2$, and $k\equiv1\pmod 4$, \endroster and when semisimple, $\roman{S}_{((ke),(je))}$ is isomorphic to $$ \bigoplus_{r=1}^{j}\roman{D}_{(((k+j-r)e,(r-1)e+1),(e-1))}\langle j\rangle \oplus \roman{D}_{((ke+je),\emptyset)}\langle j\rangle. $$ The method the authors use to achieve this relies on a Morita equivalence between an appropriate category of imaginary representations of KLR algebras and the category of representations of the classical Schur algebra constructed by A.~S. Kleshchev and the first author in [Mem. Amer. Math. Soc. {\bf 245} (2017), no.~1157, xvii+83 pp.; MR3589160]. \par Working in the setting of Schur algebras allows the authors to obtain the result in some of the non-semisimple cases, and obtain the graded decomposition for all Specht modules indexed by bihooks found before. In addition, the authors show that if ${p=2}$ then $\roman{S}_{\lambda}$ is decomposable if and only if $k\nequiv j \pmod{2^l}$, where $l$ is given by ${2^{l-1}\leqslant j<2^l}$, thus filling a gap in the results in [L. Speyer and L. Sutton, op. cit.], and thus strengthening the conjecture above.