= Solution
The multiplier enforcing <elastic incompressibility> contributes an arbitrary hydrostatic <stress>. Denote it by $q(r)$ to distinguish it from the applied <pressure> $P$. For the <neo-Hookean solid>, the <Cauchy stress tensor> is $\tau=-qI+\mu FF^T$. Thus its principal components are
$$
\boxed{t_r=-q+\mu\lambda_r^2=-q+\mu\frac{R^2}{r^2},\qquad
t_\theta=-q+\mu\lambda_\theta^2=-q+\mu\frac{r^2}{R^2}.}
$$
The axial component is $t_z=-q+\mu$, despite the unit axial stretch. The quantity $q$ is a <Lagrange multiplier> determined together with equilibrium and the <traction> conditions; it is not obtained by differentiating the reduced stretch <energy> while ignoring its constraint.
In particular the <stress> difference eliminates this multiplier:
$$
\boxed{t_\theta-t_r=\mu\left(\frac{r^2}{R^2}-\frac{R^2}{r^2}\right).}
$$
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