Measure algebra on the Bohr compactification vs the bidual algebras

I'm going to say no. The "canonical" pairing of $M(bG)$ with $L^\infty(G)$ is to integrate a function in $L^\infty(G)$ against the restriction to $G$ of a measure in $M(bG)$. But this is not faithful: any measure supported on $bG\setminus G$ would go to zero in $L^\infty(G)^*$. To pick up mass on this corona we would want to extend functions in $L^\infty(G)$ to $bG$. But you have to be almost periodic to canonically extend to $bG$, so it doesn't seem like there's any way to get what you want.