For a pointed metric space $(M,d,0)$, we denote by $Lip_0(M)$ the Banach space of all real-valued Lipschitz functions $f$ defined on $M$ and such that $f(0)=0$. Recall that $Lip_0 [0,1] = L^\infty$ (see, for example, N. Weaver's book Lipschitz Algebras). It's not hard to see that the hyperplane $H$ defined above is isometrically isomorphic to $Lip_0([0,1]/\{0,1\})$, where $[0,1]/\{0,1\}$ the metric space obtained by collapsing $\{0,1\}$ to one point (see again Weaver for quotient metric spaces), that is, $[0,1]/\{0,1\}$ is the unit circle with the length distance. $Lip_0 ([0,1]/\{0,1\})$ can be seen as the subspace of $Lip_0 [0,1]$ formed by the functions $f$ such that $f(1)=0$. Take the obvious linear Lipschitz extension operator $T$ from $Lip_0 \{0,1\}$ to $Lip_0 [0,1]$ and define $\Phi: Lip_0 \{0,1\} \oplus_\infty Lip_0 ([0,1]/\{0,1\}) \rightarrow Lip_0 [0,1]$ by $\Phi(f,g)\doteq T(f)+g$. It is straightforward that $\Phi$ is an onto isomorphism with $\|\Phi\|=2$ and $\|\Phi^{-1}\|=2$. Since $Lip_0\{0,1\} \oplus_\infty Lip_0 ([0,1]/\{0,1\})$ is isometrically isomorphic to $Lip_0 ([0,1]/\{0,1\})$, it follows that $d_{BM}(H,L^\infty)\leq 4$.

Combining with Bill Johnson's answer we then have $2\leq d_{BM}(H,L^\infty) \leq 4$, and the search for the exact value continues.

From the above construction we get an explicit isomorphism between $H$ and $L^\infty$; not a very natural one though.