The most we could do, maybe, is investigate whether the physical substrate of our minds makes them uncopyable, and therefore whether it's even logically coherent to imagine a distinction between them and copyable minds.
If that's the most you're expecting to show at the end of your research program, then I don't understand why you see it as a "hope" of avoiding the philosophical difficulties you mentioned. (I mean I have no problems with it as a scientific investigation in general, it's just that it doesn't seem to solve the problems that originally motivated you.) For example according to Nick Bostrom's Simulation Argument, most human-like minds in our universe are digital simulations run by posthumans. How do you hope to conclude that the simulations "shouldn't even be included in my reference class" if you don't hope to conclude that you, personally, are not copyable?
Scott Aaronson has a new 85 page essay up, titled "The Ghost in the Quantum Turing Machine". (Abstract here.) In Section 2.11 (Singulatarianism) he explicitly mentions Eliezer as an influence. But that's just a starting point, and he then moves in a direction that's very far from any kind of LW consensus. Among other things, he suggests that a crucial qualitative difference between a person and a digital upload is that the laws of physics prohibit making perfect copies of a person. Personally, I find the arguments completely unconvincing, but Aaronson is always thought-provoking and fun to read, and this is a good excuse to read about things like (I quote the abstract) "the No-Cloning Theorem, the measurement problem, decoherence, chaos, the arrow of time, the holographic principle, Newcomb's paradox, Boltzmann brains, algorithmic information theory, and the Common Prior Assumption". This is not just a shopping list of buzzwords, these are all important components of the author's main argument. It unfortunately still seems weak to me, but the time spent reading it is not wasted at all.