= Solution
Let $\mathbf n=\mathbf R/R$. The first <sphere>'s leading <velocity field> is $\mathbf u_0=a^3\boldsymbol\Omega_0\times\mathbf x/r^3$. Taking its <curl> gives
$$
\boldsymbol\omega_0(\mathbf x)=\frac{a^3}{r^3}\bigl[3\widehat{\mathbf x}(\widehat{\mathbf x}\cdot\boldsymbol\Omega_0)-\boldsymbol\Omega_0\bigr].
$$
The second <sphere> is <torque-free>, so
$$
\boxed{\boldsymbol\Omega_R\sim\frac{a^3}{2R^3}\bigl[3\mathbf n(\mathbf n\cdot\boldsymbol\Omega_0)-\boldsymbol\Omega_0\bigr].}
$$
Its own <force-free> and <torque-free> motion emits no <Stokeslet> or <rotlet>; further reflections are smaller at large separation. In particular it spins in the opposite direction for a primary rotation perpendicular to the line of centers, and in the same direction with twice that magnitude for a parallel rotation.
Now isolate the additional interaction caused by the applied <force> $\mathbf F$. Its leading <Stokeslet> produces at the first center the incident <velocity> $(I+\mathbf n\mathbf n)\mathbf F/(8\pi\mu R)$. The <Stokes drag law> makes the increment of the <force> required to hold that <sphere> fixed
$$
\boxed{\mathbf F_0\sim-\frac{3a}{4R}(I+\mathbf n\mathbf n)\mathbf F,\qquad |\mathbf F_0|=O(a|\mathbf F|/R).}
$$
The sign opposes the incident <velocity>. This holding <force> acts on the fluid and generates a reflected <Stokeslet>. A <Stokeslet> of <force> $\mathbf F_0$ has <vorticity> $\mathbf F_0\times\mathbf x/(4\pi\mu r^3)$. Evaluating half that <vorticity> at $\mathbf R$ gives
$$
\delta\boldsymbol\Omega_R\sim\frac{\mathbf F_0\times\mathbf n}{8\pi\mu R^2}=\boxed{\frac{3a}{32\pi\mu R^3}\mathbf n\times\mathbf F}.
$$
Thus its magnitude is $O[(|\mathbf F|/(\mu a^2))(a/R)^3]$, as required. This is <forced-sphere rotation reflected from a held sphere>. Longitudinal forcing has zero leading spin by <symmetry>. Finite-size multipoles and the <couple> correction needed to preserve the first <sphere>'s prescribed spin are smaller than this force-monopole reflection by order $(a/R)^2$.
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