Ok, then substitute "quantum state", which is more general and allows for the somewhat different mathematical objects that model photons in QFT.There's no wave function for photons!
If your criticism is that, strictly speaking, QFT has no invariant concept of "the quantum state of an extended system at an instant of time", that is true, but you can still pick a particular frame, such as the overall rest frame of the system + measurement apparatus in the experiment under discussion, and construct the quantum state of an extended system such as the multi-photon system in the experiment at an instant of time in that frame. That is implicitly what is being done when we talk about the "wave function" of the multi-photon system.
Also, as has been pointed out, you could do a similar experiment with electrons moving at non-relativistic speeds. Such an experiment could be modeled using only non-relativistic QM, in which the description in terms of wave functions is unproblematic and the issues you appear to be raising with photons not having wave functions do not arise.