"I've long thought that Marx was being ironic about this - that labor power is paid at a "just" rate is just evidence of how preposterous the notion of capitalist justice is (liberty, equality, and Bentham). The notion that the value of labor power is its cost of reproduction makes sense under bourgeois norms, for sure. Why else should we care about exploitation if it isn't unjust?"
Well he was often ironic, but he was pretty forthright on this question. In particular he mocked the 'fair day's work for a fair day's pay' on the grounds that this was the formula that would ensure, not obviate, exploitation.
I think you have to read his interest in exploitation as a critical development of the arguments of the Ricardian Socialists. He was interested in the way they used Ricardo's economics to lay claim to the full product of labour. But his differentiation of the commodity 'labour power' from labour product shows that just exchange leaves the surplus in the capitalists' hands.
When he makes exploitation the cornerstone of his analysis it is not because he thinks it unjust, it is because he thinks it is what makes the relationship dynamic - and that means that it will tend to develop to the point where it is redundant. That way he takes the moral critique (capitalism is unfair) and turns it into a materialist one (capitalism is historically redundant).
What Marx says about inside the factory, as opposed to outside, is less that it is exploitative, more that it is oppressive, because the capitalist knows no limitation upon his right to dispose of his property apart from the limits that the workers impose upon him: 'between equal rights, force decides'.
It won't do to criticise capitalism from a standpoint of justice that somehow exists outside of capitalism, Marx is saying to us, because our notions of justice are merely an idealised projection of the exchange relations that are characteristic of capitalist society. All that would achieve is the perfection of the market system.
In the Critique of the Gotha Programme:
"Do not the bourgeois assert that the present-day distribution is "fair"? And is it not, in fact, the only "fair" distribution on the basis of the present-day mode of production? Are economic relations regulated by legal conceptions, or do not, on the contrary, legal relations arise out of economic ones? "
This is why Marx is so scathing about the Gotha programme's implied commitment to equality, which he says is itself a bourgeois standard that must be left behind:
"In a higher phase of communist society -- only then then can the narrow horizon of bourgeois right be crossed in its entirety and society inscribe on its banners: From each according to his ability, to each according to his needs!"