Is it really a good idea to use electrically conducting fluids to cool a computer?
With a high enough potential difference,
everything conducts <grin>.
So long as the fluid is adequately separated from the necessary conducting elements by insulating materials, it would be fine. I'd be more worried about the magnetic fields playing fast-and-loose with the electrical current flows.
For innovative cooling methods, of you were to drill a hole through the case into the body of the radiator/heat-sink and tap it, you could screw in an extender (using thermal paste on the screw threads) which you could cool externally. Simplest would be a copper or silver dish or flask which holds ether/isopropanol/water/ice/mixture of salt and ice/dry ice/liquid nitrogen. Cooling effectiveness would probably be limited by the capacity of the heat-pipe to transfer heat from the processor to the radiator/heat-sink and cooling element. Alternatively, screw in an external heat-pipe and radiator.
I really would not want a Pyra to have a fan. Noisy, and tend to clog with dust. A designed-in interface to an optional external chiller/cooler for those that want it would be preferable, in my view. Obviously, other people have different views, which I respect.
The back of the screen/lid would be a great place to put a heat-pipe radiator as you have the possibility of good convective flow of air plus relatively large surface area, which you could increase with vertical fins. The (big) problem would be needing some flexible heat-pipe plumbing between the processor and lid. Way too complex to contemplate.