Try more basic, more conceptual, than that:
For a ~DC photon to be a reasonable concept, there must be a persistent bias or structure in the universe giving rise to it, for the better part of the age of the universe.
It also must be, not just that there was a source in the past, and will be a sink arbitrarily in the future, but that it's still present and ongoing, or at least was very recently (~billions of years).
Such a field could manifest as an electromagnetic bias in the universe. It could be as basic as the cosmological constant, or dark energy: a pressure that permeates space, not really interacting with anything directly, but indirectly through its energy density, its effect on spacetime curvature.
That does assume some sort of origin for such a phenomenon, like charge imbalance. Which is exceedingly unlikely to be the case (all observations point to a damned neutral universe).
But again, wave-particle fallacy.
The photon particle is just the quantized manifestation of the EM field. It is the Fourier transform (frequency domain) representation of the transient waveform. The definition of the FT is frequencies (sine waves) that exist for all times. There is nothing necessarily causal or realistic about frequency domain, it's just another tool we use to work with these sorts of problems. (And, as it turns out, most physical processes exhibit some sort of frequency dependency, or explicit frequency levels -- energy levels -- as a result of quantization. So it is a very useful tool indeed!)
Speaking about frequencies over time is tricky at best. It's much better to fall back to a transient model in that case. The Schroedinger equation is just a differential equation, meaning, for given boundary conditions, it has some sort of solution; and if the boundary conditions are suitable, then that solution has quantized energy levels, and therefore frequencies and wavelengths. You don't need to solve it in this way. You can solve it the same as any other difference equation, like SPICE does: by integrating a small timestep at a time. On the upside, you can solve for chaotic and non-analytic* conditions: you just keep on stepping, until you divide by zero or something!
*Though, they might be divergent, and attempts to make those divergent forms converge, may not be physically realistic. (Or maybe they'll be too realistic**.) Note that chaotic behavior is bounded, like a sine wave, and tends to be cyclical, but, the cycle rate is unbounded, so it cannot be analyzed with a mere Fourier transform.
**QED is solved analytically (and at a deeper level, not for given boundary conditions), so it's not by analogy with a transient solution. But it does encounter infinite divergent sums. These can be forced through (the series of partial sums diverges, but the infinite series can still be assigned a finite value, using renormalization), with the result being the most accurate physical theory we know to date. So, that's fun, huh?
So the takeaway point is this: photons are just another conceptual tool, like the FT. They are not a useful tool at very low frequencies. They're not inapplicable, but trying to force meaning from such a view is a stretch at best. One must always keep in mind that the physical laws must be consistent. If classical E&M suffices to explain an observed field, then one gains little insight from repeating the same exercise in a higher (quantum) domain.
Tim