= Solution
Remove the common nonzero factor $e^{-i\omega t}$ and expand in the small parameter $\epsilon$. To first order, the modified <Laplacian> contributes only its $y$ derivatives: its $x$ derivative term has the extra factor $\epsilon^2$. The <regular perturbation> expansion therefore gives
$$
\boxed{P=-i\omega b_0+\omega\psi_{0,y},\qquad Q=-i\omega b_1+\omega\psi_{1,y}+\psi_{0,x}-b_{0,yy},}
$$
$$
\boxed{R=-i\omega\psi_{0,yy}-\omega b_{0,y},\qquad S=-i\omega\psi_{1,yy}-\omega b_{1,y}-b_{0,x}.}
$$
Thus the two residuals are $P+\epsilon Q+O(\epsilon^2)$ and $R+\epsilon S+O(\epsilon^2)$, respectively. Subscripts after a comma denote <partial derivatives>. We retain $\omega\ne0$, as appropriate to the oscillatory wave and the elimination below.
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