David Kaplan uses the following example in support of the idea that indexicals and demonstratives (including some uses of pronouns) refer directly, i.e. unmediated by Fregean senses:
Suppose you have a friend, Paul, who lives in Princeton. You're at a party, and Charles has shown up disguised as Paul. You say
(1) "He [Delta] now lives in Princeton."
Kaplan says, "I assume that in the possible circumstances described earlier, Paul and Charles having disguised themselves as each other, Delta would have demonstrated Charles. Therefore, under the Fregean theory, the proposition I just expressed, Pat [he named the proposition 'Pat'] would have been false under the circumstances of the switch" (Demonstratives, 516). He construes Fregeanism for some reason as tying the actual object presented, rather than the mode in which it is presented, with the proposition expressed. (Sure, senses are modes of presentation of referents, but senses compose thoughts, so I'm a bit confused.)
He claims that Direct Reference gets the right result because demonstratives are rigid designators; they designate the same thing in all possible worlds (keeping certain things constant), rather than varying with, say, which object is present. (I can't claim to understand this. If you hold the object constant, the person you refer to in all possible worlds would still be Charles, would it not?)
But this seems like the right result for the wrong reason. I agree that in the proposition above, the person you refer to is Paul rather than Charles. But why? Because the information you're drawing on in making the judgment is derived from Paul, and not Charles. It's irrelevant whether Paul or Charles is the person sitting next to you: you're saying something about Paul based on information drawn previously from Paul.
Let's tweak the case slightly. Suppose, at the same party, Charles-in-a-Paul-suit starts dancing. You say,
(2) "I never thought I'd see him dance!"
Which person are you referring to?
This seems harder, to me, because the information you're drawing on concerns the actions of the person in front of you. Are these Charles's actions, or Paul's? I don't pretend this is obvious, but I think it's much more tempting in this case to say you're referring to the person who's actually there (Charles), rather than the person he's disguised as. Then again, it might not be surprising if Charles is someone who dances often, and Paul never does. So perhaps here you're referring to Paul also, even if Charles is doing the dancing.
What if Charles disguised himself as Paul to carry out a murder. You catch him in the act, and call the police. When they arrive, you say,
(3) What he did was horrible!
Does 'he' refer to Charles, or Paul? Charles. Why? Charles did the thing, even if you thought he was someone else. These are not easy cases.
A very Evansian way to treat these cases would be the following: you are referring to the person from whom the bulk of the information you draw on in your thought/statement/judgment derives. You fail to meet your target in making the judgment insofar as there is a mismatch between the person from whom the bulk of your information derives and the person who's actually there. So in (1) you have tried and failed to refer to Paul by demonstrating Charles, but you have still said something true about Paul. In (2) you're drawing on information about the frequency of Paul's dancing, and so still trying to refer to Paul, albeit also failing. In (3) you have said something true about Charles without knowing it. You have referred to Charles, but as it were by luck.
That is still not to explain why a more sophisticated Fregeanism might be useful in such cases. But Charles-under-a-Paul-mode of presentation could certainly go a long way in making sense of (1) and (2).