  Let $X$ be a Banach space with separable dual. It is proved that for every
$\varepsilon\in (0,1)$, $X$ embeds isometrically into a Banach space $W$ with a
shrinking basis $(w_n)$ which is $(1+ \varepsilon)$-monotone. Moreover, if $X$
has further an FDD $(E_n)$ whose strong bimonotonicity projection constant is
not larger than $\mathcal{D}$, then $(w_n)$ has strong bimonotonicity
projection constant not exceeding $\mathcal{D}(1 +\varepsilon)$. Further, if
$(E_n)$ is $\mathcal{C}$-unconditional then $(w_n)$ is $\mathcal{C}(1 +
\varepsilon)$-unconditional. The proof uses renorming and skipped blocking
decomposition techniques. As an application, we prove that every Banach space
having a shrinking $\mathcal{D}$-unconditional basis with
$\mathcal{D}<\sqrt{6}-1$, has the weak fixed point property.
