I'm not sure if there is any way to simplify $W(xa^x)$. It's pretty clear that $a=e$ simplifies to $x$ or $W_k(xe^x)$, but any other value of $a$, other than trivial values like $a=0,1$, don't seem simplifyable.
Here is a graph you may want to view. It appears to be asymptotic to a linear function is the most I can observe.
From the graph, I observed
$$W(xa^x)\sim\ln(a)x$$
This isn't an answer, though I'd really like to see one:
Well, $xa^x=xe^{x \ln(a)}$.
Consequently, if you define: $y=xe^{x \ln(a)}$, it is well defined that:
$x \ln(a)=W(y\ln(a))$
And I'm not sure if that is any more helpful. I'll definitely research a bit more though.