= Solution
Sphere 2 is now <force-free> and <torque-free>. Its translation and spin follow the incident rigid-motion components, so it emits neither a <Stokeslet> nor a <rotlet>. Its first disturbance is a <stresslet> from the incident rate of strain, of magnitude $\mu a^3(a|\mathbf V|/R^2)$. Its return <velocity> at sphere 1 is consequently $O(a^4|\mathbf V|/R^4)$:
$$
\boxed{|\mathbf U_1-\mathbf V|=O(|\mathbf V|\epsilon^4).}
$$
More precisely, that leading correction is $-(15/4)\epsilon^4(\mathbf V\cdot\mathbf n)\mathbf n$. It vanishes for purely transverse forcing, for which the first translational correction is of order $\epsilon^6$.
The first-return spin requires an additional cancellation. The leading incident strain at sphere 2 is
$$
\mathbf E=\frac{3a}{4R^2}(\mathbf V\cdot\mathbf n)(I-3\mathbf n\mathbf n).
$$
It is axisymmetric about the line of centers, so the reflected <stresslet> has zero <vorticity> at sphere 1 on that axis. A naive <vorticity> estimate of order $(|\mathbf V|/a)\epsilon^5$ therefore has zero coefficient. The finite-size correction to the incident strain and the next force-free scattering multipoles are two powers of $\epsilon$ smaller; these generically give
$$
\boxed{|\boldsymbol\Omega_1|=O\left(\frac{|\mathbf V|}{a}\epsilon^7\right).}
$$
This <suppressed spin from a force-free distant sphere> is also consistent with the reciprocal torque-to-self-translation coupling. Purely longitudinal forcing gives zero spin exactly by axial symmetry. Distinguishing the <stresslet>'s on-axis curl from its generic off-axis scale is essential here.
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