Interesting idea that you can't falsify a theory of dark matter. Never thought of that.
And I think you're right, no matter what, you could come up with some distribution of dark matter that would fit any observation.
That said, I think such distributions would come to resemble the epicycles which were once used to explain the motion of the solar system, and would be come increasingly unlikely and implausible.
I did a bit of searching, and some theories of dark matter are falsifiable and have already been falsified--such as "dark matter is X or Y" and we've shown that those cannot be.
What really racks my brain is this: some galaxies rotate like they have no dark matter. This would not be possible under MOND, unless, of course, MOND is a nonuniform-in-space theory too, which would also possibly not be falsifiable. But it's easy to hypothesize that somehow the dark matter and detectable matter got separated in this particular galaxy, explaining the rotation curve.
Really deep stuff. I have no firm belief in any particular theory at the moment, would like to see evidence.....