1. I ought to know whether the external world is actually real (axiom).
Saying that this is something we should be able to do is entirely different from saying we're morally obligated to do it. I see absolutely no moral connection to epistemology--which you seem merely to assume from the beginning.
2. If I ought to know this, I can know it (ought-implies-can in traditional ethical reasoning).
While it's true that ought implies can, this is really only useful in inferring the contrapositive case, i.e. that if we can't do something, we're not morally obligated to do it and can't be held responsible for failing to do so. It's not a way to derive "cans," it's instead a way to rule out "oughts." Because ought implies can, you can only obligate me for things that I can first do. (It's also a way to draw a distinction from the opposite inference, i.e. just because you can doesn't mean you should.)
3. If the external world is not actually real, I can't know whether it is (so far unproven assertion, but seems intuitive to me).
This doesn't seem intuitive to me. For instance, scientists say there is a way to test if the universe is a hologram. We also wake daily from our dreams. And we have lots of ways to detect and eliminate illusion.
C. Therefore, the external world is actually real.
Even if your argument was valid, logical necessity does not necessitate existence. With this conclusion, your argument has moved from ethics to ontology, not ethics to epistemology. The problem of the justification and means of our knowledge isn't the same as whether or not the world is real (though of course they are related).
I think a better argument for how we know that the external world is real is a historical account for our own being. By a close examination of the (apparent) external world, it seems highly probable that we came into being through a detailed, painstaking, evolutionary process that presupposes an external world that works by principles which link together in so many ways that the idea of it being an illusion is untenable. If the external world were an illusion, it would be one of such mind-boggling consistency, that the present moment would have a "backup illusion" going back billions of years ... an illusion of (literally) astronomically unnecessary depth. Accounting for such an illusion is an even harder problem than accounting for reality. Thus, the illusion hypothesis is the most problematic.
Since our very existence is dependent upon an explanation rooted in an external, material world, our own existence becomes a kind of justification for belief in a material world.
There is at least one sense in which it is useful to wonder whether or not we ought to (or "should be able to") know this. Indeed, there is nothing about physical creatures arising through evolution that would predispose us to expect that knowledge of our own origins was necessary or even probable. As I argue in my thread about how does evolution produce consciousness and intelligence, the ability to fathom the world to the depth that we do is entirely unnecessary from the perspective of our survival/reproductive context. Far from the expectation that we ought to be able to plumb the depths of reality, it is instead a wonder that we can at all. In other words, the very theory that accounts for our being here (i.e. arising through time in a material world)--and which therefore justifies knowledge of the material world--still has yet to explain why we're intelligent enough to be able to comprehend such a theory in the first place.
Without a crucial missing piece to this puzzle, we
ought not to be able to know the reality of the world ... and yet we do.