(1) If $A$ is $1$-complemented in its bidual, then $A$ is an AW*-algebra.

Indeed, assume that $A$ is $1$-complemented in its bidual, via a contractive projection $p\colon A^{**}\to A$. By a theorem of Tomiyama, $p$ is a conditional expectation. This implies that $A$ is monotone closed, that is, every upward directed family of self-adjoint elements in $A$ has a supremum. In particular, $A$ is an AW*-algebra. (A C*-algebra is an AW*-algebra if every maximal abelian $*$-subalgebra is monotone complete, see [1].)

(2) Every separable AW*-algebra is finite-dimensional.

Indeed, assume $A$ is a separable AW*-algebra. Consider a maximal abelian $*$-subalgebra (=masa) $B$ of $A$. Then $B\cong C(X)$ for some compact, Hausdorff, extremally disconnected space $X$ (such spaces are also called Stonean). Every metrizable, extremally disconnected space is discrete. Therefore, $X$ is finite. Thus, every masa in $A$ is finite-dimensional. This implies that $A$ is finite-dimensional. (Every infinite-dimensional C*-algebra contains a positive element with infinite spectrum, and hence an infinite-dimensional masa.)

Combining (1) and (2) we obtain that a separable, infinite-dimensional C*-algebra is never $1$-complemented in its bidual.

[1] arxiv.org/abs/1501.02434