= Solution
Put $K=\sqrt\gamma(a\otimes dB^\dagger-a^\dagger\otimes dB)$, with the time argument suppressed. Since the incoming field commutes with the adapted cavity operators,
$$
K^2=\gamma\{a^2\otimes(dB^\dagger)^2-aa^\dagger\otimes dB^\dagger dB-a^\dagger a\otimes dB\,dB^\dagger+(a^\dagger)^2\otimes(dB)^2\}.
$$
Applying the <Gaussian quantum noise Ito table> gives
$$
K^2=-\gamma\,dt\,Q\otimes I_B,\qquad Q=Naa^\dagger+(N+1)a^\dagger a-M(a^\dagger)^2-M^*a^2.
$$
The <power series> expansion $e^K=I+K+K^2/2+\cdots$ consequently becomes, through Ito order $dt$,
$$
\boxed{U_I=I+\sqrt\gamma(a\otimes dB^\dagger-a^\dagger\otimes dB)-\frac\gamma2Q\otimes I_B\,dt.}
$$
This has precisely the required quadratic drift correction. The short-time ordering is essential: the linear noise term is order $\sqrt{\gamma dt}$ and the drift is order $\gamma dt$; higher stochastic orders are discarded in the differential limit. A remainder written merely as a power of $\gamma$ is a formal coupling expansion, not a dimensionally complete statement of the short-time error.
As a sign check, $K^\dagger=-K$ and $Q^\dagger=Q$. Thus $U_I^\dagger U_I=I-\gamma Qdt-K^2=I$ through this order. The cross term is needed for stochastic <unitarity>.
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