I will attempt a qualitative answer. In your last comment you state:
It looks like I'm asking whether details of the micron-to-millimeter scale geometry of the bunches affect relative cross-sections for different processes in nontrivial ways (whereas the only number usually exchanged between beam physicists and detector physicists is the relatively crude measure of luminosity, which in any case is superfluous insofar as there is a count of the total number of events
The mm and micron scale is orders of magnitude larger than the scale where the interactions of interest, protons on protons for the sake of the argument, can have a measurable effect, order of fermi and less , 10^-15m. The goal is to have a high energy proton hitting an opposite high energy proton and measure the apparent cross sectional area. The difference in scales is such that any modulation in the shape of the beams cannot play a role except in the gross probability of beam meeting beam, and this is what the luminosity supplied takes care of. BTW luminosity is necessary to be provided because just counting the number of interactions would not give us a cross section, i.e. area, measurement to relate to calculated crossections from the theories. It is a kind of calibration.
Another way of looking at this is to suppose that the proton crossection is a fixed number , protons are little billiard balls. The number of deviations from the beam direction of the classical scatter, given the luminosity , would give us this fixed crossection. Any beam modulation would reflect in the rates per unit time and it will be only the errors in the measurement that would be affected by the modulations.