= Solution
Use the four-parameter ansatz from part (ii), now allowing every parameter to vary slowly. Oddness of $y f_E(y)^2$ and the stationary-profile integrals give the <collective-coordinate effective Lagrangian>
$$
\boxed{L_{\rm eff}
=N(E)(\dot\theta-u\dot X)
+H_0(E)+\frac12N(E)u^2
+\epsilon N(E)V(\epsilon X)+O(\epsilon^3)},
$$
where
$$
N(E)=2\sqrt{2|E|},
\qquad
H_0(E)=-\frac{[2|E|]^{3/2}}3,
\qquad
\frac{dH_0}{dE}=E\frac{dN}{dE}.
$$
Variation of $\theta$ makes $N(E)$, and hence $E$, constant. Variation of $u$, $X$, and $E$ then gives
$$
\boxed{\dot X=u,
\qquad
\dot u=-\epsilon^2V'(\epsilon X),
\qquad
\dot\theta=-E+\frac12u^2-\epsilon V(\epsilon X)}.
$$
Thus the center follows the classical Hamiltonian
$$
\boxed{H_{\rm center}(X,u)=\frac12u^2+\epsilon V(\epsilon X)}
$$
to leading adiabatic order. The first neglected profile correction is of order $\epsilon^3$, because the term linear in $y$ integrates to zero.
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