= Solution
The <angular momentum> <commutators> used in the frame transformation are representation-independent. Therefore $H_R=\Delta S_z+\Omega_RS_x$ still holds. Rotating its axis onto $z$ gives $2s+1$ <eigenvalues> and exact laboratory solutions
$$
\boxed{E_\mu=\hbar\mu\Omega,\quad
|\Psi_\mu(t)\rangle=e^{-i\mu\Omega t}R(t)|\mu\rangle_R,\quad
\mu=-s,-s+1,\ldots,s}.
$$
Now $R(T)=(-1)^{2s}I$, so $e^{i\Phi_\mu}=(-1)^{2s}e^{-i\mu\Omega T}$. In the adiabatic limit the <dynamical phase> is $-\mu BT$ and
$$
\boxed{\gamma_\mu=-\mu\mathcal A\pmod{2\pi}}.
$$
Indeed $2\pi s+2\pi\mu\cos\theta$ and $-2\pi\mu(1-\cos\theta)$ differ by $2\pi(s+\mu)$, an integer multiple of $2\pi$. Integer <spins> return with a positive rotation sign and half-integer <spins> with a negative one; the <geometric phase> need not vanish for either.
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