First of all, $\int_0^t\exp[B(t)-B(s)]\,dW(s)$ should be understood as $\exp[B(t)]\int_0^t\exp[-B(s)]\,dW(s)$, otherwise, the integral couldn't be defined as an Ito's integral.
Let \begin{align} Z(t)&\Bigl(=\int_0^t\exp[B(t)-B(s)]\,ds\int_0^t\exp[B(t)-B(s)]\,dW(s)\Bigr)\\ &=e^{2B(t)}\int_0^t\exp[-B(s)]\,ds\int_0^t\exp[-B(s)]\,dW(s) \end{align} Using Ito's formula, we have \begin{align} dZ(t)&=2Z(t)\,dB(t)+2Z(t)\,dt+\Bigl(e^{B(t)}\int_0^t\exp[-B(s)]\,dW(s)\Bigr)\,dt\\ &\quad +\Bigl(e^{B(t)}\int_0^t\exp[-B(s)]\,ds\Bigr)\,dW(t) +2\gamma \Bigl(e^{B(t)}\int_0^t\exp[-B(s)]\,ds\Bigr)\,dt \tag{1} \end{align} Denote $m(t)=\mathsf{E}[Z(t)]$, then $m(0)=0$ and from (1) we have \begin{align} dm(t)&=2m(t)\,dt+\mathsf{E}\Bigl[e^{B(t)}\int_0^t e^{-B(s)}\,dW(s)\Bigr] +2\gamma\int_0^t\mathsf{E}[e^{B(t)-B(s)}]\,dt\\ &=2m(t)\,dt+6\gamma(e^{t/2}-1).\tag{2} \end{align} where we use following equalities: \begin{align} \mathsf{E}\Bigl[e^{B(t)}\int_0^t e^{-B(s)}\,dW(s)\Bigr]&=2\gamma(e^{t/2}-1),\tag{3}\\ \mathsf{E}[e^{B(t)-B(s)}]&=e^{(t-s)/2} \end{align} ToNow from (2) and $m(0)=0$ we could get the expression of \begin{align} m(t)&=\mathsf{E}\Bigl[\int_0^t\exp[B(t)-B(s)]\,ds\int_0^t\exp[B(t)-B(s)]\,dW(s)\Bigr]\\ &=\gamma e^{2t}-4\gamma e^{t/2}+3\gamma.\end{align}
To get (3) using same way as above(Ito's formula and solving a ODE).
Now fromLet (2)$X(t)=e^{B(t)}\int_0^te^{B(s)}\,dW(s)$ and $m(0)=0$ we could get the expression of $$m(t)=\mathsf{E}\Bigl[\int_0^t\exp[B(t)-B(s)]\,ds\int_0^t\exp[B(t)-B(s)]\,dW(s)\Bigr].$$$m_X(t)=\mathsf{E}[X(t)]$, then
\begin{align}
dX(t)&=X(t)\,dB(t)+\frac12X(t)\,dt+dW(t)+\gamma dt,\\
dm_X(t)&=\frac12m_X(t)+\gamma dt, \qquad m_X(0)=0\\
m_x(t)&=\gamma\int_0^te^{(t-s)/2}\,ds=2\gamma(e^{t/2}-1).
\end{align}
Please excuse me couldn't give the final expressionthat above deduction may be included some errors and typos, since the process includes
many many details be checked.