LePore and Cumming aren't ideally clear, perhaps. But the point they are making is actually a straightforward one, and fairly clearly correct too.
Suppose a claim A or B is made which is naturally understood, in the context, as telling us overall that either $A$ or $B$ but not both.
Does this give us evidence that "or" has an exclusive meaning in English (alongside the inclusive meaning it undoubtedly has)?
Well plainly not, if there is another explanation of our reading in this case, that is compatible with "or" having an uniquely inclusive meaning.
And that will be the situation, at least in a range of examples.
In particular, suppose that $A$ and $B$ are already logically incompatible. Then in this case, of course the disjuncts of A or B can't be true together -- and if we know this, then we will understand A or B as conveying the overall message that $A$ or $B$ but not both. In other words, in this sort of case, we have no reason to suppose that our exclusive reading is due to a special exclusive meaning of "or", when it can simply be due to our understanding that $A$ rules out $B$.
So, take the example "Today is Monday or today is Tuesday". Ok, we hear this as conveying an exclusive message, let's agree. But the disjuncts are incompatible, and we know them to be so. So we can explain why we read the overall message in that claim to be that only one of the disjuncts holds without supposing that that is part of the meaning of "or" itself.
More generally -- and this is LePore and Cumming's correct point -- disjunctions A or B with evidently incompatible disjuncts aren't good examples to use to try to show that ordinary-language "or" itself has a special exclusive meaning, since in these cases we have another explanation of why we construe the overall message as exclusive. (But their mode of presentation of this correct point, I certainly agree, is pretty poor.)