= Solution
For a static equilibrium, <linearization> of magnetic induction gives
$$
\delta\mathbf B=\nabla\times(\boldsymbol\xi\times\mathbf B_0)
=(\mathbf B_0\cdot\nabla)\boldsymbol\xi-(\boldsymbol\xi\cdot\nabla)\mathbf B_0-\mathbf B_0\nabla\cdot\boldsymbol\xi.
$$
The rotating local basis must be differentiated. In particular, $\partial_\parallel\hat{\mathbf x}=\mathbf b_0/R$ and $\partial_\parallel\mathbf b_0=-\hat{\mathbf x}/R$. Their contributions proportional to $\xi_\parallel$ cancel in the radial component, yielding
$$
\boxed{\delta b_x=ik_\parallel\xi_x,\qquad\delta b_y=ik_\parallel\xi_y.}
$$
Writing $h=\delta B/B_0$, the magnitude component is
$$
h=ik_\parallel\xi_\parallel+\frac{\xi_x}{R}+\frac{\xi_x}{L_B}-\nabla\cdot\boldsymbol\xi
=\frac{\xi_x}{L_B}-ik_y\xi_y,
$$
where $\nabla\cdot\boldsymbol\xi=\xi_x/R+ik_y\xi_y+ik_\parallel\xi_\parallel$ when $\partial_x\xi_x=0$.
Set $M=B_0^2/(4\pi)=\rho_0v_A^2$ and $\delta\Pi=\delta p_\perp+Mh$. The transverse $y$ component of the linear <kinetic magnetohydrodynamic momentum equation> is
$$
\rho_0(\omega^2-k_\parallel^2v_A^2)\xi_y=ik_y\delta\Pi.
$$
Since $k_y\gg k_\parallel\sim\omega/v_A$, perpendicular <magnetic pressure> enforces the leading <pressure-balanced plasma perturbation> condition $\delta\Pi=0$, giving
$$
\boxed{h=-\frac{4\pi\delta p_\perp}{B_0^2}=-\frac\beta2\frac{\delta p_\perp}{p_0}.}
$$
In the radial component, the curvature part of the perturbed magnetic force is $-2Mh/R$: one contribution comes from $(\mathbf B_0\cdot\nabla)\delta\mathbf B$ and one from $(\delta\mathbf B\cdot\nabla)\mathbf B_0$. The perturbed <pressure anisotropy> contributes $-(\delta p_\perp-\delta p_\parallel)/R$. After the leading total-<pressure> constraint has removed $\partial_x\delta\Pi$, the radial equation is
$$
-\rho_0\omega^2\xi_x=-Mk_\parallel^2\xi_x-\frac{2Mh}{R}-\frac{\delta p_\perp-\delta p_\parallel}{R}.
$$
Substituting $Mh=-\delta p_\perp$ gives
$$
\boxed{(\omega^2-k_\parallel^2v_A^2)\xi_x=-\frac{\delta p_\perp+\delta p_\parallel}{\rho_0R}.}
$$
Although $h$ is small at low <plasma beta>, its contribution $Mh$ to the <force balance> is of the same order as $\delta p_\perp$ and must be kept.
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