maybe the real worry about counterpart theory is something like this?

John Divers' Possible Worlds has a nice discussion of the worry that counterpart theory doesn't adequately justify the extent to which we are ego-concerned with our own possibilities. If the possible Humphrey that won the election is a distinct creature in a universe not spatially connected to ours, what does that matter to the actual Humphrey, very much concerned with the possibility of his own winning or losing?

Divers does a very good job on behalf of the counterpart theorist in trying to undermine this worry. It mostly involves showing how non-philosophical sentences involving ego concern end up coming out true as interpreted by the counterpart theorist. It's a little bit weak in that properties we might take to be intrinsic end up being relational. This is only a weakness because Lewis and Divers take this kind of thing to be a criticism of the person who holds that worlds overlap (the same Humphrey existing at multiple worlds), and this is why Divers himself only considers counterpart theorists who believe in the reality of non-actual possible worlds, and actualists who don't. But if you have to explain putatively intrinsic things relationally, why not do it to avoid counterparts in the first place? I think for Lewis the other part of the puzzle is a horror at ontic vagueness, which the overlapper would be more likely to face. For Lewis the possible worlds and objects aren't vague, but there is vagueness in our decision to take certain objects to be counterparts or not.

I'm still not up to date on this literature, but I think that Divers at least doesn't present the best argument to justify Kripke's original worry about ego concern. This is clear if we consider duplicates instead of counterparts. Duplicates are objects existing in the same world that could serve as counterparts if they existed in different worlds.* Given the kinds of recombination principles that Lewis and Divers countenance, it should follow that for any two counterparts at different worlds, there is a world where objects indiscernible to both counterparts (as well as the environmental aspects that make them work as counterparts) exist.

Now lets consider the world where there is a United States1 where Humphrey1 loses the election and a United States2 where Humphrey2 wins it. Let's say that Humphrey1 doesn't know about United States2, but that it occurs to him that for all he knows there's an actual mirror United States not that different from his own, but where he wins the election. Then let's say at some point in the future he develops a good enough intra-world telescope to see United States2 and Humphrey2 being president. He might indeed infer from  that it was possible that he won the election.

But he wouldn't identify Humphrey2's winning the election with the possibility that he won. He would be right not to. Most analogical reasoning works in exactly the same way. Objects in the actual world are perceived to have some property and it is inferred that objects that are relevantly similar have the same property. These are often, if not usually, modally rich properties. If a certain medicine works on a certain kind of primate, we infer that it will probably work on us. But we don't identify the drug's actually curing primates with it's potential to cure us. Even weaker kinds of ontological dependence seem to fail here. A's being evidence for B does not mean that A grounds B.

Assuming that this is correct, what difference does it make whether United States2 is spatio-temporally connected to United States1? Again, we want to say that what actually happens in hypothetical worlds might be evidence for what might happen here. But if being evidence for and being the truth-condition of are two different relations in the actual world, why should they be collapsed when we are talking about two different worlds? I just can't see that it's a difference that makes a difference.

I think that the overlapper is on stronger grounds here, since there is a difference between a duplicate doing something in this world and what you yourself do in another possible world.

[*This is an example what Divers calls an "extraordinary context" where we are talking about the modal status of the set of possible worlds itself. For the genuine realist all such sentences are, if true, vacuously necessarily true. I've got a student working on this issue, and will blog about it in the next week or so.]