The answer is no.
Let $X$ be a separable Pisier counterexample [P] to Grothendieck’s problem. That is, both $X$ and $X^*$ have cotype 2 and every operator from $X$ or $X^*$ into a Hilbert space is 2-absolutely summing. As Pisier points out, there is a constant $C$ so that if $T$ is a finite rank operator from $X$ (or $X^*$) into a cotype 2 space, then $\pi_2(T) \le C\|T\|$, where $\pi_2(\cdot)$ is the 2-summing norm. This implies that if a finite rank operator $T$ from $X$ into a cotype two space satisfies $\|Tx\| \ge \delta \|x\|$ for all $x$ in some n-dimensional subspace of $X$, then $\|T\| \ge \delta c n^{1/2}$, where $c>0$ depends only on the cotype 2 constant of the range of $T$. Pisier spaces are weird, to be sure, but Pisier proved that every separable cotype 2 space isometrically embeds into a separable Pisier space.
Let $(E_n)$ be an increasing sequence of subspaces of $X$ with $E_n$ having dimension $n$ so that $\cup_{n} E_n$ is dense in $X$, and let $Y$ be the $\ell_1$ sum of $(E_n)$. There is a natural quotient mapping $Q$ from $Y$ onto $X$: given $e_n \in E_n$ with $\sum_n \|e_n\| < \infty$, set $Q(e_n)_n : = \sum _n e_n$. This quotient mapping has local liftings, which implies that $Q^*$ is an isometric embedding of $X^*$ onto a norm one complemented subspace of $Y^*$. Consequently, $X$ embeds isometrically into $Y^{**}$. (Details of this argument can be found in [J]). The nice thing about $Y$ is that it has cotype 2 since $X$ does, and $Y$ obviously also has the metric approximation property (even a monotone finite dimensional decomposition).
Claim: The weak$^*$ operator closure of $B(X,Y)$ in $B(X,Y^{**})$ has empty interior.
To see the claim, take any isometric embedding $J$ from $X$ into $Y^{**}$. Take any $n$ dimensional subspace of $JX$; for definiteness we use $JE_n$. Norm $JE_n$ up to $1+\epsilon$ by finitely many unit functionals in $Y^*$. If the claim is false, then there is an operator $J_n: X \to Y$ s.t. $\|J_nx\| \ge (1+\epsilon)^{-1} \|x\| $ for all $x\in E_n$, and the norms $\|J_n\|$ are uniformly bounded (the bound depending on the size of the ball that is contained in the weak$^*$ operator closure of $B(X,Y)$ in $B(X,Y^{**})$). Since $Y$ has the metric approximation property, $J_n$ can be taken of finite rank. But since $Y$ has cotype 2, Pisier’s result forces $\|J_n\| \ge c n^{1/2}$ for some constant $c>0$.
So see that the OP’s question has a negative answer, consider the space $Y \oplus X$.
[J] Johnson, William B. A complementary universal conjugate Banach space and its relation to the approximation problem. Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972). Israel J. Math. 13 (1972), 301–310 (1973).
[P] Pisier, Gilles Counterexamples to a conjecture of Grothendieck. Acta Math. 151 (1983), no. 3-4, 181–208.