First of all, the importance of the diffeomorphism on points in configuration space is that the cotangent lift (see 6.3 of Cotangent Bundles) $T^*f$ is guaranteed to be symplectic. If $M$ is a cotangent bundle $T^*Q$ then the obvious choice for the pre quantum bundle $B$ is $M \times\mathbb{C}$. Indeed in the prequantization we introduce the polarization by requiring that $q$ goes to $\frac {\partial} {\partial p}$
A symplectic manifold (M,ω) is said to be prequantizable if the
integral of $\omega$ around any 2-cycle lies in $2\pi\mathbb{Z}$
Now given the Lie Algebra $\mathfrak{h}$ of a compact and connected Lie group $H$, the character group $\hat H$ can be identified with $H^1(H,\mathbb{Z})$. The Maslov class of a lagrangian immersion $\iota: L → T^∗M$ is defined as the degree-1 cohomology class and determines via the exponential map $\mathbb{R} \to U(1)$ an isomorphism class of flat hermitian line bundles over $L$ (see pag. 56 of the pdf linked in the question).
Another way of looking at this is that if we impose the strong Legendre condition, that $\mathscr L$ is globally a diffeomorphism, then Lagrange’s equations on $TQ$ are transfromed by $\mathscr L$ into Hamilton’s equations on $T^∗Q$ with canonical symplectic structure.
Finally, given a Lagrangian submanifold $\Lambda$ of the cotangent bundle, contained in the energy shell $H^{-1}(E)$, it is desirable to attach the WKB expansion to the manifold $\Lambda$ (see Semi-classical approximations)
The WKB method applies when the semi-classical density in phase space
is supported by a Lagrangian submanifold: for the levels of a
separable system this leads to the Bohr-Sommerfeld rules corrected by
the Maslov index.
And that corresponds to the Maslov quantization condition, described in the already mentioned Lectures on the Geometry of Quantization at pag. 45
Returning to the harmonic oscillator of Example 4.5, we see that the
level set $H^{−1}(E)$ satisfies the Maslov condition provided that for
some integer n, $E = (n + 1/2)$. Allowable energy levels in this case
therefore correspond to the Bohr-Sommerfeld condition, which actually
gives the precise energy levels for the quantum harmonic oscillator.