= Solution
Let the tangential <velocity> scale be $U$, gap scale $H$ and along-gap length $L\gg H$. The tangential Stokes equation balances $P/L$ with $\mu U/H^2$, while wall shear is $\tau\sim\mu U/H$. Therefore
$$
\boxed{P\sim\frac{\mu UL}{H^2},\qquad \frac\tau P\sim\frac HL\ll1.}
$$
This is <lubrication pressure dominates shear stress>; <pressure> <force> projections may nevertheless be small in a particular direction or may give no <torque> on a spherical surface.
Write the signed centre displacement along the downward axis as $d=\alpha\Delta$. A point $an$ on the inner sphere has distance $|an+d e_z|=a+d\cos\theta+O(\Delta^2/a)$ from the outer centre. The leading normal gap to radius $a+\Delta$ is consequently
$$
\boxed{h(\theta)=\Delta-d\cos\theta=\Delta(1-\alpha\cos\theta).}
$$
The local-curvature corrections are smaller by $\Delta/a$. This approximation also follows by solving the outer-sphere intersection along the ray $n$ from the inner centre and expanding its positive root.
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