Is there any hope of a resolution along these lines: GR will hold true (or will have held true) once we make a local measurement of all particle positions, and between now and then there is still a well-defined spacetime whose points have a well-defined conformal structure (or light-cone structure, or "causal" structure), just not a well-defined metric structure"?
I don't think this works, because it treats the problem as if measurement had some special status in quantum mechanics. This is not true in standard quantum mechanics, i.e., in the set of axioms of QM that everyone agrees on. Measurement only gets any special status if you add additional axioms that talk about measurement, as in the Copenhagen interpretation.
I don't want to pooh-pooh your idea that we should seriously poke and prod ideas like measurement. Measurement is foundational in QM, but the formalism we have for QM never really refers to measurement. If I use QM to calculate the photon-emission rate for a hydrogen atom, none of the equations have any observers or measurements in them. So you would think the same would be true for something like the graviton-emission rate, if we kept something like the same formalism. But we clearly need to make one or more foundational changes to either QM or GR in order to get a theory of quantum gravity, and people like Lee Smolin have indeed suggested that such a change might have something to do with the role of observers and measurement. The heuristic argument is that we want to be able to talk about things like a wavefunction that represents a cosmological spacetime, but there is no external observer who can do measurements on such a wavefunction. I just don't think anybody knows for sure whether such a foundational change is what's needed or what it would look like, and I don't think it would be minor tinkering. It would be a major modification, because our current formalism of QM so completely sidesteps these issues, other than just having certain assumptions like the existence of an inner product (which you could argue is a way of talking about measurement and probabilities without actually talking about them).
Putting aside the specific issue of measurement, more broadly I think what you're proposing is to couple a quantum-mechanical system (matter) to a classical one (the gravitational field), but there are both general arguments (dating back to 1927) and arguments specific to this case that one cannot in general do this type of coupling. An example of the latter is the Eppley-Hannah argument (described in Adelman, cf. Carlip). An easy 1927-era argument, about coupling a classical EM field to quantized matter, is that it forces you to discard conservation of energy and momentum, except statistically, because you can't have the quantum-mechanical long-range correlations that prevent, e.g., the same photon from being absorbed by two atoms. This was the motivation for the experimental work by Bothe and Geiger that disproved the BKS theory. To avoid this type of problem, you really need to quantize the gravitational field, so then we have to have things like superpositions of different metrics. But nobody knows how to make sense of this, and it would certainly preclude the idea of well-defined light cones, even in the sort of restricted sense you seem to be suggesting.
I don't know, it may be possible to do what you're talking about as a semiclassical approximation scheme. The standard prescription for semiclassical gravity is that in the Einstein field equations, you replace the stress-energy tensor with its expectation value. But this can't be fundamentally right, and actually semiclassical gravity seems to be a horrific mess. They can't predict anything, even in contexts where you would think a semiclassical approximation might work, e.g., the vacuum state near the event horizon of a black hole, which should look like a vacuum anywhere else.
Adelman, "The Necessity of Quantizing Gravity," http://arxiv.org/abs/1510.07195
Carlip, "Is Quantum Gravity Necessary?," http://arxiv.org/abs/0803.3456
