= Solution
The geometry and boundary velocities are axisymmetric and linear in the axial pseudovector $\boldsymbol\Omega$. The pressure is a true scalar. No nonzero parity-even scalar can be formed linearly from $\boldsymbol\Omega$ and the radial vector $\mathbf x$: $\boldsymbol\Omega\cdot\mathbf x$ is a pseudoscalar. Hence the flow-induced pressure is spatially constant.
Equivalently, taking the divergence of the Stokes momentum equation gives the <Harmonic pressure and vorticity in Stokes flow> result
$$
\nabla^2p=0,
$$
and symmetry plus the homogeneous boundary data excludes every nonconstant harmonic pressure mode. Since pressure is defined up to an additive constant, we choose
$$
\boxed{p=0}.
$$
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