Wave function constrained to a fixed trajectory? Really? this is probably a very stupid question. First of all, I'm a mathematician so please try to use coordinate-free notations.
It's often used in quantum mechanics a wave function depending on a fixed trajectory. For instance, in How to derive the Aharonov-Bohm effect result?, $\psi (x, \gamma)$ depends on the path $\gamma$. I can understand this in terms path integral quantization (which is essentially perturbative and I don't like). However in the philosophical scope of quantum mechanics it seems unreasonable to fix a trajectory unless the wave function is collapsed at all possible times (we know the position at every instant). Note that the holonomy $\exp (-iq_e \int A)$ is gauge invariant, but $A$ (that's actually a cocyle $A =\{A_i, {g_{ij}}\}$ representing a gauge equivalence) need not to be $d$-closed (i.e., $dA_i = F_A$ may be non-zero), therefore it really depends on the path chosen (and not just on the endpoints). Maybe this condition is simply implicitly posed on the boundary conditions? 
Let me be more precise. I will treat the Aharonov-Bohm effect in order to deal with something more concrete. For simplicity, let's take the space-time $X = \mathbb{R} \times \mathbb{R}^3$ together with a charge at $(t,0)$, i.e., $$dF_A = q_m\delta_{(t, 0)}$$ and a particle of charge $q_e$ moving through  fixed paths $\gamma_i$, $i = 1, 2$ with the same endpoints.
What's the meaning of $\psi (x, \gamma_i)$ in this case? Is the wave function really collapsed at every instant (i.e., do we always know the position)?
If the above questions cannot be answered without propagators or path integrals, then is the Aharonov-Bohm effect a strictly perturbative phenomenum? If no, how to derive the Aharonov-Bohm effect non-perturbatively?
By deriving the Aharonov-Bohm effect I mean obtaining a phase difference that would lead to the Dirac charge quantization condition. For instance, in the path integral formulation the phase difference is given by $\exp (-iq_e \int_{W_e} A)$ (where $W_e$ is a loop through the solenoid) and this leads to $q_m q_e \in 2\pi \mathbb{Z}$ if one wants the solenoid to be undetectable.
For sake of completeness, I will add a possible solution of the Dirac charge quantization condition that does not makes the Aharonov-Bohm effect visible in any meaningful way, so that maybe someone can include the Aharonov-Bohm effect in this context.
A possible solution (to the Dirac charge quantization and not to the Aharonov-Bohm effect) is given at page 14-16 of http://www.maths.ed.ac.uk/~jmf/Teaching/Lectures/EDC.pdf , by declaring that $\chi = \log (g)$ for some gauge transformation $g$, that is the two open charts $U_{i} = \{(x, y, z) \in \mathbb{R}^3| \ (-1)^i z > 0 \}$ covering $\mathbb{R}^3\setminus 0$ together with the potentials $A_{-}$ and $A_{+}$ defines a $U(1)$-bundle. 
Thanks in advance.
 A: Consider the following setup for electromagnetic field:
$$\phi(x,y)=U_0\theta(R^2-x^2-y^2),$$
$$\vec A(x,y)=\frac{A_0}2\begin{cases}
\left(y,-x,0\right)&x^2+y^2\le R^2\\
\left(\frac{R^2y}{x^2+y^2},-\frac{R^2x}{x^2+y^2},0\right)&\mathrm{otherwise}
\end{cases},$$
where $\theta(x)$ is Heaviside theta, and $U_0$ is assumed to be very large — such that the electron would never get inside (may be infinity, i.e. homogeneous Dirichlet boundary condition at $x^2+y^2=R^2$), and $R$ is the radius of a solenoid.
If you propagate an electron wave so that it would go in the direction of the solenoid, the result of the interference on the side opposite to the source would depend on $A_0$, even though $\vec B=\nabla\times\vec A$ vanishes whenever $x^2+y^2>R^2$, and the wavefunction is nonzero only there. This is the essence of the Aharonov-Bohm effect. All the talk about pairs of fixed paths in this topic is merely a simplification to aid analysis.
A: Would like to add to some previous answers.
The Klein-Gordon / Schrodinger's equation masquerades as a wave equation in time and space, but it is actually a statement about frequency and wavevector.
In principle the 'wave' equation in a constant magnetic field and electron beam source can be symmetrized and solved semiclassically or by a mix of Born and partial-wave expansion techniques. Judging by your apparent math comfort level, I will describe a technique you can implement easily.
You are not measuring phase difference in path. You are measuring a curve of arrival phase by field strength, relative to the phase of arrival when no field is present. It just so happens that on one side of the solenoid the phase is wave is stretched and on the other it is compressed. 
The simplest way to calculate the total phase and amplitude differenes is to pick two points opposite your solenoid, $(x_i,y_i)$ and $(x_f,y_f)$ that form a line through your solenoid, plus an arbitrary point in between, $(x',y')$ on the dividing plane through the solenoid whose normal is the line you just chose. You need to calculate both the transverse momentum dispersion affect on amplitude and also the phase shift which occurs on the route between your three points, which is two propagations (plane waves). Once you complete this, you can integrate over all $(x',y')$ points in that plane (or a section of the plane) to find your complete phasor at the final point. This weighted sum of triangle paths well-approximates the actual curved wavefront trajectories of the disturbances in the true electron phasor field.
This procedure has nothing to do with quantumness. Multiple propagation is the same procedure you use when solving Fresnel-Kirchoff optics wave problems.
Finally. Addressing your question about the regularity of the Berry curvature field. Just like the metric in GR can become singular, there are also singular points for the Berry curvature. And worse, when we generalize the problem beyond amplitudes. However, most phenomena do not require that level of inspection. Operationally the effect acts only on the 4-momentum eigenstate spectrum, which is barely connected to its dual in spacetime at all in the statistical limit. 
There is actually a classical analogue known as the fast-slow adiabatic cycling drag. (See J. Hannay 1985). His mathematics is easy to understand and compare to Berry's paper on QM (1984). 
