This isn't anything near a proof that we don't live inside a black hole, but it's a bundle of evidence that certainly goes against it, and that the Great Attractor is not, in fact, the singularity.
First off: Expansion of the universe.
As you no doubt know, the universe is expanding. In fact, the expansion is accelerating. Do black holes expand? Yes. As they suck in more matter, they can get bigger. But if that was the case, we should notice more matter coming into the universe (well, I suppose it could be from outside the visible universe, but we should still see lots of matter coming towards us). Also, the universe's expansion is driven by dark energy, and it pushes (on a large scale) everything away from each other. In an expanding black hole, there would be no reason for the matter inside the event horizon to move away from each other; only the event horizon expands.
You also made a good point in a comment below about Hawking radiation. Eventually, in the far future, when there is nothing left in the universe but black holes, black holes will evaporate via Hawking radiation (okay, they do that now, but they can still take in more matter). If our universe is a black hole, it should then contract. But we see no reason why it should. In fact, the theory that predicts the universe's eventual collapse into a singularity (i.e. the opposite of the Big Bang), the Big Crunch theory, predicts that the universe's contraction will match its expansion. Contraction due to Hawking radiation wouldn't necessarily mirror the black hole's growth. Also, the Big Crunch theory is supported only by a minority of scientists due to the evidence against it.
Second off: The motion of galaxies by the Great Attractor.
First off, see When will the Milky Way "arrive" at the Great Attractor, and what all happen then? (and not just my answer! I dearly wish @LCD3 would expand his/her comment into an answer!). The general gist of things there, as pertains to your remark about the Great Attractor, is that the galaxies aren't all moving towards it. There are doubts (see the papers I mentioned) that the galaxies previously thought to be moving towards it are, in fact, moving towards more distant objects - other superclusters. If the Great Attractor was indeed the singularity, a) all the galaxies in the universe should be accelerating towards it, which is not the case, and b) we should be moving directly towards it, and due to its gravity, not that of the superclusters beyond.
Like I said in my comments, I don't quite think your last part relates to your first part, but I'll try to address it. First, I'm not sure where you got your sources, but I can say that we don't really know what goes on inside a black hole, and I don't now how someone got that density figure (I am, of course, not the authority on black holes - see @JohnRennie for that, on Physics SE, and I could be wrong about this). The density inside the event horizon would not be very thin, however, in regions where there is lots of matter. For example, in a black hole with an accretion disk, the material that is being absorbed might not have that low a density. Also, a large volume of that density would not necessarily collapse to become a black hole, because it would not be compact enough.
Hope it helps.