(I'll be drawing upon the responses I've previously made to Blar and Rex Hubbard when they offered these objections.)
First, note that the skeptic's challenge is whether we can have actual knowledge in light of the counterfactual possibility of our being BIVs. The question is not whether our knowledge would survive were we actually massively deceived -- obviously it would not. False beliefs cannot constitute knowledge. So the only cases worth considering are ones where we have true beliefs.
The skeptic accepts this, but argues that even if we have true beliefs about the external world, these cannot constitute knowledge because our subjective uncertainties mean that our beliefs lack justification. The externalist response is to deny that knowledge requires subjective awareness of justification. Instead, they suggest, our beliefs must be reliable or 'justified' in a more objective sense. There must be an appropriate connection between the truth of the belief and our believing of it. What matters for knowledge is that this connection holds as a matter of fact. It does not matter whether we are aware of the connection.
To reiterate: the skeptic is willing to grant that my belief that "I am not a BIV" is true. So the externalist is not begging the question when he assumes that I'm not a BIV. (Besides, we can always discharge the assumption via conditional introduction, if you insist. That is, we conclude that if I am in fact not a BIV then I in fact have knowledge. Since it's certainly possible that I'm not a BIV, it follows that we can, possibly, have knowledge.) And the externalist will certainly agree that we cannot show that we're not BIVs. He grants the skeptic this much. The dispute is simply over whether I can know that I am not a BIV, given that my belief is true but undemonstrable. On an externalist conception of knowledge, I can have such knowledge.
(Of course, that doesn't show that externalism is the true theory of knowledge. Perhaps it isn't. But that's another issue. The current question is whether the truth of externalism would defeat skepticism. And the answer is clear: it would.)
Another side-issue is whether it is in fact true that these objective conditions required for knowledge hold. For example, is it true that I'm not a BIV? I obviously believe so, but this isn't something I can show. But the whole point of this post is to explain why this doesn't matter. I can't show that I have knowledge, because I can't show that I satisfy the objective conditions required for knowledge (which at a minimum include truth). But, contrary to the skeptic, this doesn't mean that I lack knowledge. A belief might be true even if we can't show it. Similarly, it might constitute knowledge, even if we can't show it.
Anyway, this all ties in rather nicely with a brilliant anti-skeptical move by Thomas Reid, recently quoted on the common sense philosophy blog:
Reason, says the skeptic, is the only judge of truth, and you ought to throw off every opinion and every belief that is not grounded on reason. Why, Sir, should I believe the faculty of reason more than that of perception; they came both out of the same shop, and were made by the same artist; and if he puts one piece of false ware into my hands, what should hinder him from putting another.
In other words, shouldn't the consistent skeptic also doubt the dictates of logic, trusting nothing — not even the conclusions of their own arguments? Such an uber-skeptic would have no chance of showing that we lack knowledge. At best, he might insist that we can't show that we have knowledge (though Reid would presumably question his certainty even of this). But even if we conclude that we can't show anything at all, that still leaves upon the possibility that we know all sorts of things. Because, as already explained, knowing does not require showing. (The latter might be sufficient for knowledge, but it isn't necessary.)
(Aside, I must say there seems something incredibly odd about being skeptical of the rational faculty. I've suggested such a skeptical scenario before, but I don't know quite what to make of it all. I guess it just goes to show how misguided is the Cartesian ideal of casting off "all assumptions" — as if it would still be possible to think at all after doing so! Hmph. Actually, this ties in with the stuff in my previous post about the need for innate constraints, the impossibility of extreme empiricism, etc. Anyway, I'll stop rambling now. Maybe the comments will shed more light on the matter...)