Via my colleague Garth Dales, some observations which answer your question in the negative, even in the abelian case:$\newcommand{\N}{{\mathbb N}}$

> We know that $K$ is hyper-Stonean iff $C(K)$ is isometrically dual. So you are asking for locally compact spaces $K$ such that $C_0(K)$  is isomorphically dual, but not isometrically a dual space.

> The easiest example is to look at $\beta\N$  and choose a point $p\in \beta\N \setminus\N$, and consider the maximal ideal
$M_p$ of functions that vanish at $p$.   This is isomorphic to $\ell^\infty$, but $M_p= C_0(\beta\N \setminus \{p\})$ and $\beta\N \setminus \{p\}$  is not even compact.

> Probably you want a compact space $K$ with this property. In our book we give a compact $K$ such that $C(K)$ is isomorphically dual, but $K$ is not even Stonean. (It is totally disconnected.)

> ... The standard example is $K=G_I$, the Gleason cover of the unit interval. $K$ is an infinite, separable Stonean space without isolated points and $C(K)$ is  isomorphically a bidual space because $C(K)$ is isomorphic to $\ell^\infty$. But $K$ is not hyper-Stonean.

The book he refers to, co-authored with Dashiell and Lau, is in production and will be published by Springer (eventually).