The second Piola Stress for incompressible hyperelastic material is expressed as
\begin{equation}
\mathbf{S}=2 \frac{\partial W(\mathbf{C})}{\partial C}+pJ\mathbf{C}^{-1}=\mathbf{S}^{'}+pJ\mathbf{C}^{-1},
\end{equation}
where $\mathbf{C}=\mathbf{F}^\textrm{T}\mathbf{F}$ and $J^2=\textrm{det}(\mathbf{C})$. In order that $p$ can be interpreted as hydrostatic pressure, the following equation has to be true, which is
\begin{equation}
p'=\frac{1}{3}J^{-1}\mathbf{S}^{'}:\mathbf{C}=0,
\end{equation}
since $J>0$, in paricular $J=1$, thus
\begin{equation}
\mathbf{S}^{'}:\mathbf{C}=2\frac{\partial W(\mathbf{C})}{\partial C}:\mathbf{C}=0.
\end{equation}
That condition is, in general, true for any valid material model if we restrict our attention only to distortional right Cauchy–Green tensor, i.e. when constrain of incompressibility is somehow imposed on deformation of the material. However, if that is not the case, $W(\mathbf{C})$, i.e. $\mathbf{C}$ is arbitrary, the free energy function has to have an appropriate form, i.e., has to be homogenous of order zero, i.e., $W(\mathbf{C})=W(\alpha\mathbf{C})$ (see note below).
Above is the source of confusion in the question, when we calculate Second Piola Stress, we have to use only the distortional component of the right Cauchy–Green tensor,
\begin{equation}
\hat{\mathbf{C}}=\textrm{det}(\mathbf{C})^{-1/3}\mathbf{C}.
\end{equation}
Equivalently we can modify free energy such that is a homogenous function of $\mathbf{C}$. That is done by introducing $\hat{W}(\mathbf{C})$, that is
\begin{equation}
\hat{W}(\mathbf{C})=W(\hat{\mathbf{C}}).
\end{equation}
For case of such homogenous function can be showen that $\hat{W} (\alpha\mathbf{C})=\hat{W}(\mathbf{C})$, see [1], thus $\hat{p}^{'}=0$, and $p$ is hydrsotatic pressure.
Applying that reasoning to the case of incompressible Neo-Hookean material,
\begin{equation}
\hat{W}=\frac{\mu}{2}(\hat{\mathbf{C}}:\mathbf{I}-3)
\end{equation}
and carefully calulating derivatives,
\begin{equation}
\frac{\partial (\hat{\mathbf{C}}:\mathbf{I})}{\partial \mathbf{C}}
=
\textrm{det}(\mathbf{C})^{-1/3}\mathbf{I}
-
\frac{1}{3}\textrm{det}(\mathbf{C})^{-1/3-1}\textrm{det}(\mathbf{C})\mathbf{C}^{-1}(\mathbf{C}:\mathbf{I})
\\=
\textrm{det}(\mathbf{C})^{-1/3}
\left(
\mathbf{I}-\frac{1}{3}(\mathbf{C}:\mathbf{I})\mathbf{C}^{-1}
\right),
\end{equation}
finally we get
\begin{equation}
\hat{\mathbf{S}}^{'}
=
\mu\left[
\textrm{det}(\mathbf{C})^{-1/3}
(\mathbf{I}-\frac{1}{3}(\mathbf{C}:I)\mathbf{C}^{-1})
\right],
\end{equation}
which for zero strain would give zero stress. Having zero stress at zero strain in prinicple is matter of definition of material reference configuration as a stress free, however is not necessity. However, proving that $p^{'}=0$, for $W(\hat{\mathbf{C}})$ is essential for validating hyperelastic model as incompressible.
Note: If we impose incompressibility constrains explicitly of deformation by Lagrange multiplier, e.g. some form of Taylor-Hood elements in case of finite elements discretization, you can use $W(\mathbf{C})$, not $\hat{W}(\mathbf{C})$, since constraints are enforced explicitly on right Cauchy-Green tensor.
Note: For homogenous function (see [2]), we have
\begin{equation}
W(\alpha\mathbf{C})=\alpha^n W(\mathbf{C}),
\end{equation}
differntatin both side by $\alpha$, we get
\begin{equation}
\frac{\partial W(\mathbf{C})}{\partial C}:\mathbf{C} = n\alpha^{n-1}W(\mathbf{C}).
\end{equation}
Thus for zero oder, when $n=0$, that imples $\frac{\partial W(\mathbf{C})}{\partial \mathbf{C}}:\mathbf{C}=0$, i.e. $p^{'}=0$.
[1] Bonet, Javier, and Richard D. Wood. Nonlinear continuum mechanics for finite element analysis. Cambridge university press, 1997.
\begin{equation}
\hat{W}(\alpha\mathbf{C})=
\hat{W}(\textrm{det}(\alpha\mathbf{C})^{-1/3}\alpha\mathbf{C})
\\=
\hat{W}((\alpha^{3}\textrm{det}(\mathbf{C}))^{-1/3}\alpha\mathbf{C})
\\=
\hat{W}((\textrm{det}(\mathbf{C}))^{-1/3}\mathbf{C})
\\=
\hat{W}(\mathbf{C}).
\end{equation}
[2] https://en.wikipedia.org/wiki/Homogeneous_function
