Comparative Statics on Balanced Growth Path
The approach is not correct, because $\mu^{BGP}$ is a result of underlying structural parameters. So to say "when $\mu^{BGP}$ increases..." immediately begs the question why it increases, which underlying parameter(s) has changed to cause such an increase... assume it was $\epsilon$ that decreased. But in this case $g_k$ is not affected at all, so you see that you cannot claim what you claim about  $\mu^{BGP}$ affecting positively $g_k$.

What is meaningful to do is to perform comparative statics for the exogenous parameters, and record whether $\mu^{BGP}$ and $g_k$ move in the same or the opposite direction (or don't move) in each case. It may not be a causal link, but co-movement is also a useful result, and testable.