But if you have a continuous distribution of momenta that can be absorbed, then the momentum change in the electron is continuous.
You mean, the possible momentum changes? Then yes.
There's no way to tell if you've absorbed four photons of one frequency, five of another, or some mix.
I don't think that's the case. For one thing, you could use something more sophisticated than an electron - something with some structure, that's more sensitive to some frequencies than others. For another you can vary the field profile and strength, and do many experiments. And in reality you have to allow some input from theory to interpret the results, and the theory predicts a certain spectrum of momentum changes due to photon absorption, and you can go out and check the predictions of the theory.
All you will be probing is the magnetic field strength. Which is not enough to determine how many photons are in the field.
Not just the strength, because even classically the profile of the field (say the volume where it's non-zero) affects the momentum transferred. But yes, the strength is part of it as well.
I'm saying that the only way to determine how many photons make up the field is to measure the field strength AND size, and then take the Fourier transform, because there's no way to directly measure those photons.
So you agree (?) I have a theory in which I can assign a finite number of photons to the field configuration, and you agree that every single prediction of that theory - calculated using discrete interactions with finite numbers of photons - is correct, but you don't agree there's a way to measure that number or be sure it's accurate. If that's your position, I can't argue with it (nor am I very interested in trying).
The FT is a non-local property of the field. But the measurement you propose is essentially a local measurement.
Why? I can shoot electrons through from any angle, through the whole thing, through part of it, etc. How is that local?
I don't think you can extract the number of photons in the field from the measurement, because multiple scenarios will produce the same measured result.
Well, there is a significant subtlety here regarding infrared divergences. Any time you have massless particles like photons, the number of "soft" photons - photons with very low momentum and energy - gets hard to define. But that's a problem that's solved the minute you consider a detector with finite energy resolution, it goes away if the universe were huge but finite (which tells you right away it can't matter), and it's there even when the field is precisely zero. So I don't think it prevents one from answering this question, although I'll agree my argument is somewhat vulnerable on that point.
As an example more topical to this thread - if I asked you how many photons there are in the sun, or how many it emits per second, would you agree that's a sensible question with a finite answer?