Not an answer.

$h$ is already universal enveloping provided only that $K$ is the rationalization of a simply connected space; this is, for example, Theorem 21.5 in Felix, Halperin, and Thomas. In particular, $h$ is injective so the answer to the two questions is the same.

As for the actual question, for starters, there is a Lie model of $K$ (which is in particular a dg Lie algebra whose homology Lie algebra is $\pi_{\bullet}(\Omega K)$) whose underlying graded Lie algebra is freely generated by an element in each degree $i - 1$ for each $i$-cell of $K$, with differentials determined by the attaching maps. This is a corollary of Theorem 24.7 in Felix, Halperin, and Thomas. Conversely, given such a dg Lie algebra $L$ we can write down a finite CW complex which has it as a Lie model. So the question reduces to the following purely algebraic question:

Does a connected dg Lie algebra $L$ whose underlying graded Lie algebra is free on finitely many generators have a finitely generated homology Lie algebra?

I imagine someone knows the answer to the analogous question for connected dg algebras; that would be helpful as a guide.