First expand
\begin{aligned}
n(\psi ; \omega)=& \exp \left(\frac{-K \sigma+\omega \psi+K \sigma \cos \psi}{D}\right) n(0 ; \omega) \\
& \times\left\{1+\frac{\left(e^{-2 \pi \omega / D}-1\right) \int_{0}^{\phi} e^{(-\omega \phi-K \sigma \cos \phi) / D} d \phi}{\int_{0}^{2 \pi} e^{(-\omega \phi-K \sigma \cos \phi) / D} d \phi} \right\}
\end{aligned}
to fourth order in $K$ and evaluate
$$\int_0^{2\pi}n(\psi,\omega)\cos(\psi)\,d\psi=\pi  (K\sigma/D) n(0,\omega)\frac{ 1+(\omega/D)^2-K\sigma/D}{\left[(\omega/D)^2+1\right]^2}+{\cal O}(K^4).$$
(Note a typo in the equation in the OP, the first exponent on the right-hand-side should contain $\psi$ instead of $\phi$.)

The function $n(0,\omega)$ is determined by the normalisation
$$\int_0^{2\pi}n(\psi,\omega)\,d\psi=1,$$
which gives to fourth order in $K$ the equation
$$n(0,\omega)=\frac{(10-(K\sigma/D-2) K\sigma/D) (\omega/D)^2+2 (K\sigma/D+2)^2+2 (\omega/D)^4}{4 \pi  \left((\omega/D)^2+1\right) \left((\omega/D)^2+4\right)}+{\cal O}(K^4).$$
We thus arrive at
$$\int_0^{2\pi}n(\psi,\omega)\cos(\psi)\,d\psi=\tfrac{1}{2}(K\sigma/D)\frac{1}{1+\omega^2/D^2}-\tfrac{1}{4}(K\sigma/D)^3\frac{6+(\omega/D)^2+(\omega/D)^4}{(1+\omega^2/D^2)^3(4+\omega^2/D^2)}+{\cal O}(K^4).$$
This then gives the desired expansion of the function $g$.