It doesn't seem to produce any sense of Doom, though, which seems to count against the horcrux hypothesis.

In canon Harry's sense of pain when encountering Voldemort doesn't occur when encountering horcruxes. Moreover, it turns out that Harry is an accidental horcrux and he doesn't have any similar reaction to himself, or even to a time traveled version of himself that he sees. So by analogy a horcrux here may not be enough to trigger the sense of doom.