Suppose $L$ is a complete lattice, $P$ is a poset, and $f: L \to P$ is a surjective order-preserving map. If ${\bf DM}(P)$ is the Dedekind MacNeille completion of $P$, is there necessarily a surjective order-preserving map ${\bar f}: L\to {\bf DM}(P)$?

No.

Take a poset $P$ on the ground set $S=\{0,1,a,b,c,d\}$ determied by $a>c$, $a>d$, $b>c$, and $b>d$ (with $0$ the minimal and $1$ the maximal element). Let $L$ be a complete lattice on $S$ with $\{a,b,c,d\}$ being an antichain (and the same minimum and maximum). Then the identical map on $S$ is an order preserving map $L\to P$, but $P$ is not a lattice, so its completion contains more than 6 elements. THus there is no sujcection from $L$ to the completion at all.