= Solution
Let $\mathbf e=\mathbf X/R$, $s=\mathbf e\cdot\mathbf B\mathbf e$, and $A_\parallel=\mathbf A\cdot\mathbf e$. Applied <forces> and couples below are <forces> and <torques> on the solid; by <force> balance their values are also the strengths exerted by that solid on the fluid. This fixes the signs in <Faxén's first law> and <Faxén's rotational law>. To hold the passive <sphere> fixed, set its translation and <angular velocity> to zero in those laws. The incident <two-mode tensorial squirmer flow> gives
$$
\mathbf F_h=-6\pi\mu a\left(\mathbf u(\mathbf X)+\frac{a^2}{6}\nabla^2\mathbf u(\mathbf X)\right),\qquad
\mathbf G_h=-4\pi\mu a^3\boldsymbol\omega(\mathbf X).
$$
Only the <stresslet> part has <vorticity>:
$$
\boldsymbol\omega(\mathbf x)=3a^2\frac{(\mathbf B\mathbf x)\times\mathbf x}{r^5}.
$$
Consequently, for fixed $\mathbf A,\mathbf B$ as $a/R\to0$,
$$
\boxed{\begin{aligned}\mathbf F_h&=-\frac{9\pi\mu a^3}{R^2}s\mathbf e
-\frac{2\pi\mu a^4}{R^3}(\mathbf A-3A_\parallel\mathbf e)
\\&\quad+O\!\left(\frac{\mu a^5\|\mathbf B\|}{R^4}+\frac{\mu a^8|\mathbf A|}{R^7}\right),\\
\mathbf G_h&=-\frac{12\pi\mu a^5}{R^5}(\mathbf B\mathbf X)\times\mathbf X+\text{higher reflections}.\end{aligned}}
$$
The leading <force> is the radial <stresslet> <force>. Its next correction, $O(\mu a^4|\mathbf A|/R^3)$, comes from the incident <potential dipole>. The next $\mathbf B$ correction, $O(\mu a^5\|\mathbf B\|/R^4)$, contains both the swimmer's $r^{-4}$ field and the <Laplacian> term in <Faxén's first law>. If the relevant coefficient vanishes, these next terms must be retained instead of calling the vanished term a nonzero leading approximation. Further exchanges in the <method of reflections for Stokes flow> are smaller still.
The passive <sphere> reflects primarily a <Stokeslet> with strength $\mathbf F_h$. At the swimmer, its <velocity> is $(\mathbf I+\mathbf e\mathbf e)\mathbf F_h/(8\pi\mu R)$. The unchanged <surface slip velocity> adds the free swimming <velocity> to the incident-flow contribution in <Faxén's first law>. Thus
$$
\boxed{\begin{aligned}\delta\mathbf U&=-\frac{9a^3}{4R^3}s\mathbf e
-\frac{a^4}{4R^4}(\mathbf A-5A_\parallel\mathbf e)
\\&\quad+O\!\left(\frac{a^5\|\mathbf B\|}{R^5}+\frac{a^6|\mathbf A|}{R^6}\right).\end{aligned}}
$$
The leading change is $O(a^3\|\mathbf B\|/R^3)$; its next correction is the displayed $O(a^4|\mathbf A|/R^4)$ term. Finite-radius <Faxén's first law> corrections, the passive <sphere>'s reflected <stresslet>, and its <rotlet> contribute at order $a^5\|\mathbf B\|/R^5$ to translation.
For a <Stokeslet> at the origin,
$$
\mathbf u_F(\mathbf x)=\frac1{8\pi\mu}\left(\frac{\mathbf F}{r}+\frac{(\mathbf F\cdot\mathbf x)\mathbf x}{r^3}\right).
$$
The <curl> of the first term is $\mathbf F\times\mathbf x/r^3$; that of the second is also $\mathbf F\times\mathbf x/r^3$, before multiplying by $1/(8\pi\mu)$. Hence $\boldsymbol\omega_F=\mathbf F\times\mathbf x/(4\pi\mu r^3)$. At the swimmer use displacement $-\mathbf X$. The leading $\mathbf B$ <force> is parallel to $\mathbf X$, so its <Stokeslet> has zero <vorticity> there. This eliminates the putative $O(a^3\|\mathbf B\|/R^4)$ rotation. The <potential dipole> itself has no <vorticity>, but the holding <force> it induces is not radial. Applying <Faxén's rotational law> to its reflected <Stokeslet> yields
$$
\boxed{\boldsymbol\Omega=\frac12\frac{\mathbf F_{h,A}\times(-\mathbf X)}{4\pi\mu R^3}
=\frac{a^4}{4R^6}\mathbf A\times\mathbf X
+O\!\left(\frac{a^5\|\mathbf B\|}{R^6}+\frac{a^6|\mathbf A|}{R^7}\right).}
$$
The passive <sphere>'s <rotlet> and reflected <stresslet> can affect that last order, so they cannot restore the larger missing rotation. When $\mathbf A\times\mathbf X=0$, the boxed coefficient is zero and the higher-order terms determine any rotation.
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