The first paragraph of the question seems irrelevant to me. Plain old classical GR allows naked singularities. It just makes it difficult to produce one by gravitational collapse, from generic initial conditions, without exotic matter.
In terms of classical GR, Hawking proved an area theorem (Hawking 1973) that says that when two or more black holes merge, the area of the event horizon always increases, assuming the weak energy condition. Hawking, Carter, and Bardeen realized that they could form a set of laws that were exactly analogous to the laws of thermodynamics, and in this system of analogies, the area of an event horizon was the analog of entropy. In terms of tested physical theories, this is the only justification for associating an entropy with the area of the event horizon. The Hawking area theorem doesn't include contributions from the singularities, so that's the justification for not including them in the definition of entropy.
However, the area theorem requires the assumption of "a regular predictable space," which is defined as one that is "strongly future asymptotically predictable" from a partial Cauchy surface (and also satisfies some topological conditions). The whole reason that cosmic censorship is of such great theoretical interest is that if it's violated, then causality is violated, in the sense that we don't have uniqueness and existence of solutions to initial-value problems, i.e., Cauchy problems. So although I haven't dug carefully into the technical details, it sounds to me like a naked singularity violates the assumptions of the Hawking area theorem.
Therefore there appears to be no way, based on tested physical theories, to discuss the entropy of a naked singularity. I don't think it should be a big surprise that entropy fails to be a well-defined concept in a spacetime that isn't causally well-behaved. For example, GR allows spacetimes that have closed timelike curves or that aren't even time-orientable, and in such a spacetime we obviously can't have any sensible analog of the second law. Also, causality breaks down in the case of a naked singularity, as expressed by John Earman's famous observation that it would be consistent with the laws of physics if we imagine that "all sorts of nasty things -- TV sets showing Nixon's 'Checkers' speech, green slime, Japanese horror movie monsters, etc. -- emerge helter-skelter from the singularity." How do you count up the entropy of the green slime that a naked singularity is intending to output? Obviously you can't.
Hawking and Ellis, The large scale structure of space-time, 1973, Proposition 9.2.7, p. 318
