Essentially, what would be the most efficient way of causing the Earth's core to cool as much as possible without damaging human civilization in any other way (the outcome can damage it, but not the process), given our current understanding of physics? Assume financial and political obstacles are not an issue.
I suspect that cooling the Earth's core significantly without destroying civilization may not be possible.
Roughly speaking, the core (inner and outer) is about half the mass of the planet (so 3x10^24kg). Assuming a specific heat capacity of 500J/kg, and a temperature change of 1000K, that gives us:
500x1000x3x10^24J = 1.5x10^30 Joules to get rid of. That's a lot.
The oceans weigh about 1.4*10^21Kg, anf would therefore require roughly
4200x100x1.4x10^21J =6x10^26J to raise to boiling point.
So you are dealing with an amount of energy sufficient to raise the oceans to boiling point about 2000 times over. Doing it once would be catastrophic for life on the planet, obviously.
So, no matter how you get the heat out of the Earth - and I can't think of any even semi-plausible mechanisms to do so* - the chances are that you'd destroy everything on the surface by doing so. Even dumping the heat near to the Earth would probably cause overheating by radiation.
Conduction: Put many, many rods of come highly conductive material all the way to the core. Except that mantle convection would quickly shear them apart, and any density contrast with the mantle would cause them to collapse.
Convection: Stir the mantle more vigorously to allow more cooling? Would disintegrate the crust and everyone on it.
Radiation : Expose the core directly to space by digging a very big hole (not physically plausible). Place a black hole at the Earth's core, and somehow stop it swallowing matter, just radiation. Would work, but probably destroy the planet, again.
Utterly insane method: nested heat pumps.
Use a geothermal heat pump to power another geothermal heat pump, which is used to directly cool the core, or whatever depth of the mantle you can reach. Localised cooling and compression would crush the pipes within weeks, if not days, but with large enough pumps (and enough pumps), you could probably set off some major earthquakes on the surface.
Of course, that leaves the issue of what to do with all that waste heat...