You cannot derive it "directly" from a Klein-Gordon equation, or from a Klein-Gordon Lagrangian.
Starting from a Klein-Gordon equation for $\psi$, and defining $\psi(\vec x,t) = e^{-imt} \tilde \psi(\vec x,t)$ ($2.103$), you get a new equation for $\tilde \psi$, which is not a Klein-Gordon equation :
$$ \ddot {\tilde \psi} - 2 im \dot {\tilde \psi} - \nabla^2\tilde \psi = 0 \tag{2.104}$$
By Fourier transform , this is equivalent to the condition :
$$ (E'^2 + 2mE'-\vec p^2) = 0 \tag{1}$$
What does that mean?
We begin with a Klein-Gordon equation for $\psi$, which, by Fourier Transform, is equivalent to the condition
$$ (E^2 -\vec p^2 - m^2) = 0 \tag{2}$$
Now, the transformation $\psi(\vec x,t) = e^{-imt} \tilde \psi(\vec x,t)$ $(2.103)$, gives the link between $E'$ and $E$, this is $E' = E - m$, this is a shift in the definition of the energy.
So, from $(2)$, we have simply : $((E'+m)^2-\vec p^2 - m^2)=0$, which is just the condition $(1)$
Now, if we suppose $|\vec p| \ll m$, this means $|E-m| \ll m $ (with $E \sim m)$, that is $E'\ll m$, so $E'^2 \ll mE'$.
Turning back to the equation $(2.104)$, which is not a Klein-Gordon equation, we see, by Fourier Transform, that we can neglect the first term relatively to the second term, and finally, you get :
$$ i \dot {\tilde \psi}= - \frac{1}{2m}\nabla^2\tilde \psi = 0 \tag{2.105}$$
About lagrangians, you will have, I think, a problem, if you want to define a lagrangian giving $(2.104)$, with only a real scalar field $\tilde \psi$, because of the $\dot {\tilde \psi}$ term.
So, you have to consider a complex scalar field, and in the Lagrangian, you will have terms like $ \dot{ \bar {\tilde \psi}} \dot{ {\tilde \psi}}$ and $im \bar {\tilde \psi}\dot{ {\tilde \psi}}, im \dot{\bar {\tilde \psi}}{ {\tilde \psi}}$, and the first term, in the approximation we discussed, is negligible relatively to the other terms.
Best Answer
You certainly couldn't recover quantum effects with a classical treatment of that Lagrangian. If you wanted to recover quantum mechanics from the field Lagrangian you've written, you could either restrict your focus to the single particle sector of Fock space or consider a worldline treatment. To read more about the latter, look up Siegel's online QFT book "Fields" [hep-th/9912205] or Strassler's "Field Theory without Feynman Diagrams" [hep-ph/9205205] for applications of the technique.