Possible solution to a major problem.

Allowing me to do this (left: original; right: modified reconstruction).

Given image i, such that i = a + n + c, where a is the astronomical signal, n is the random noise, and c is the detector specific corruption, define a new image i' = (a - a') + n + c, where we've removed an estimate of the astronomical signal a'.  This has a Fourier transform F(i') = (A - A') + N + C, using the fact that Fourier transforms are linear.  Assuming a' is a sufficiently suitable proxy for a, and that n is small (specifically that ||N|| << ||F(i')||), this leaves F(i') ~ C.  Constructing a Fourier mask that removes C results in t ~ (a - a') + n + c.  The difference of i' and t provides a clean estimate of c, which can then be removed from i: o = i - (i' - t).

I need to test that photometric properties are retained by this transformation, and I'd like to speed it up some and sort out some of the edge case issues (visible around the blank bands above), but I think this solves the bulk of the issues with this problem.