Swap $x$ and $k$ in GGM construction
$F'_k(x)$ is not necessarily pseudo-random.

Proof:
Changing one output does not affect PRGness, so if there is a PRG

that stretches by a factor of 2 then there is a PRG that stretches

by a factor of 2 and sends all-zero strings to all-zero strings.

For all $n$ and $k$ and $x$, if $G$ is such a PRG then $F'_k(x)$ is the string of $n$ zeros.

For all $n$, if $\: n\neq 0 \:$ then there are other strings with that length.

Therefore $F'_k(x)$ is not necessarily pseudo-random.

(Of course, this doesn't prove that your construction can't be pseudo-random.)