The following post is an edit of a previous one which was not accurate.
It provides a partial answer to the question.
As a compromise I decided to include also a bit of a generalization.
Thanks to Cameron Zwarich for spotting a gap in my previous post.
Thanks to Nik Weaver for suggesting a way around it using a stronger assumption and to Cameron Zwarich for identifying this assumption with the established property of having a "strongly unique predual". That was very helpful.

Recall that the Banach space $X$ is called a *unique predual* if for every isometric isomorphism $\phi:X^*\simeq Y^*$ there exists an isometric isomorphism $\phi_*:Y\simeq X$. The Banach space $X$ is called a *strongly unique predual* if moreover there exists such $\phi_*$ satisfying $\phi=(\phi_*)^*$.
The latter notion seems more natural.

Note that the existence of a Banach space which is unique predual but not strongly unique predual seems to be an open problem - at least it was in 1989 - see X.2 in "Existence and uniqueness of isometric preduals: a survey" by Godefroy (*Contemp. Math*, 131-193, **85**, 1989).
In particular, it seems that the spaces $\text{Lip}(Y)$ which motivated Nik's question are known (by him) to be strongly unique preduals, thus my answer below, though partial, does help in proving that for a bounded metric space $X$ the space $\text{Lip}_0(X)$ has a strongly unique predual.
This is mentioned as an open problem in Nik's book "Lipschitz Algebras" (p. 40).

**Theorem:** Assume $E$ has a *strongly unique predual* and $E_0<E$ is a weak*-closed subspace such that $E/E_0$ is *reflexive*.
Then also $E_0$ has a *strongly unique predual*.

The following is a compactness criterion for preduality, due to Dixmier, which I learned from the answer by Cameron Zwarich.
For a proof see also Proposition VIII.1 in the above mentioned Godefroy's survey.

**Dixmier's Compactness Criterion (DCC):** For a Banach space $X$ and a subspace $Z<X^*$, the closed unit ball of $X$ is Hausdorff compact with respect to the weak topology generated by $Z$ iff
the restriction map $X^{**}\to Z^*$ restricts to an isometric isomorphism $X\simeq Z^*$ on $X<X^{**}$.

**Lemma:** If $X$ is a strongly unique predual and $Z<X^{**}$ is a subspace such that the closed unit ball of $X^*$ is Hausdorff compact with respect to the weak topology generated by $Z$ then $Z$ coincides with the canonical image of $X$ in $X^{**}$.

*Proof of the lemma:* By DCC the map $\phi$ given by the composition $X^*\hookrightarrow X^{***}\to Z^*$ is an isometric isomorphism. Since $X$ is a strongly unique predual, we can find $\phi_*:Z\simeq X$ such that $\phi=(\phi_*)^*$. $\phi^*:Z^{**}\to X^{**}$ restricted to $Z<Z^{**}$ gives the inclusion. Its image coincides with the image of $X$, by the commutativity of the diagram
$\require{AMScd}$
\begin{CD}
Z @>\phi_*>> X\\
@V V V @VV V\\
Z^{**} @>\phi^*>> X^{**}
\end{CD}

*Proof of the theorem:*
Let $E_*$ be the unique predual of $E$. Let $E_0<E$ be a weak*-closed subspace and assume $E/E_0$ is reflexive.
Let
$L=E_0^\perp<E^*$. By the weak* closedness assumption on $E_0$, upon identifying $E_*$ with its embedding in $E^*$, $L<E_*$.
We identify $L$ with $(E/E_0)^*$ and, by reflexivity, we identify $E/E_0$ with $L^*$.
We identify $E_0$ with $(E_*/L)^*$ and argue to show that
$E_*/L$ is a strongly unique predual of $E_0$.

Assume having an isometric isomorphism $\phi:E_0 \simeq F^*$.
Let $\phi_*$ be the isometry
$F \to F^{**} \simeq E_0^*$
and let $\pi$ be the restriction map $E^*\to E^*/L\simeq E_0^*$.
By Banach-Alaoglu the closed unit ball of $E_0$ is Hausdorff compact under the weak topology generated by $\phi_*(F)$.
It follows that the intersection of the closed unit ball of $E$ with $E_0$ is Hausdorff compact under the weak topology on $E$ generated by $\pi^{-1}(\phi_*(F))$.

We claim that also the closed unit ball of $E$ itself is Hausdorff compact under the topology generated by $\pi^{-1}(\phi_*(F))$.
To see this we establish a converging subnet for each net $x_\alpha$ in this ball. Denoting by $\bar{x}_\alpha$ its image in $E/E_0$, using the fact that $L<\pi^{-1}(\phi(F))$ generates a Haudorff compact topology on the closed unit ball of $E/E_0$ by reflexivity, we may and do pass to a subnet and assume that $\bar{x}_\alpha$ converges to $\bar{y}\in E/E_0$ for some $y\in E$.
By the Open Mapping Theorem there is a net $z_\alpha\in E_0$ with $\|(x_\alpha-y)-z_\alpha\|\to 0$. By the already established compactness of the closed unit ball in $E_0$, we may and do pass to a subnet and assume that $z_\alpha$ converges to $z\in E_0$. It follows that $x_\alpha$ converges to $y+z$.
This proves the claim.

Applying the lemma with $X=E_*$ and $Z=\pi^{-1}(\phi_*(F))<E^*$
we conclude that
$\pi^{-1}(\phi_*(F))=E_*$ inside $E^*$.
Thus, $\phi_*(F)=\pi(\pi^{-1}(\phi_*(F)))=E_*/L$,
and by the definition of $\phi_*$ we get that $\phi=(\phi_*)^*$.