I can only comment on the basic problem: photon scattering off definite field conditions.
As an example here is a lowest order scattering of a photon with an electric field, represented by virtual photons,
The on mass shell photon enters on the top left and leaves on the top right. The diagram will give the probability of scattering when calculated and the electric field value is used.
When hitting the electric field of a lattice, whether transparent or not, the field is a boundary condition for the scattering of the photon.
All boundary conditions in this sense are instantaneous, otherwise one could not do calculations. There is no velocity of light involved , except if the field is changing, when the field can only change within the limits of velocity of light.
If your memory of Feynman's statement is correct, the answer is that the information of the thickness of the lattice, on which the photon impinges, is already embedded in the topology of the electric and magnetic fields of the lattice on with which the photon is interacting. If there is some change in the lattice, that information has to travel with the velocity of light for changes to embed in the field.
Edit in order to make clear with this simple experiment the difference between the probability nature of the photon wave function, and the energy in space electromagnetic wave functions . It also demonstrates the existence of the fringe fields of matter on which the photon scatters.
Here is an experiment one photon at a time:
Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.
The experiment is: single photon at a time , of given energy, scattering.
The boundary conditions are: two slits of a given width, a tiny distance apart
On the left each photon footprint shows up as a dot in the (x,y) of the screen, (the (z is the distance of the slits to the screen), time is not recorded. There is nothing wavy of the footprint, it looks like a classical particle footprint hitting a plane.
The photons directions look random.
As one progresses from left to right and more photons are accumulated an interference pattern slowly appears and on the far right we see the expected and well defined mathematically classical interference pattern of double slit interference.
The experiment shows the probability wave nature of same energy and same boundary condition photons, and also of how the classical electromagnetic wave emerges from the seemingly random quantum behavior.
This is because the quantum wavefunction is modeled by a quantized Maxwell's equation. How the classical fields develop from the quantum substrate is seen here .
The boundary conditions for the photon solution are given by the electric field around the two slits, on/through which the photon scatters as it goes through. These conditions are embedded, the way the thickness of a lattice is embedded in the field the photon scatters off. The information that it is two slits of given width and distance is there, whether there are photons impinging or not.
The same is true for a scattering from a lattice, the information of its thickness is embedded in the fringe field topology off which the photon scatters in the question above.