You have answered my question as to whether or not you know of someone who is trying to explain the non-viscous states of matter seen recently at BNL and by NASA trying to explain the behavior of a spinning high mass neutron star. The answer appears to be no. I hope I can provide something of an answer to your questions.
Thinking classically, although I know that this may be inappropriate, I see that with helium superfluids, both He-4 (bosonic) and He-3 (fermionic) repel themselves after London forces are switched off due to extreme cooling. This possible repulsive force is of the same magnitude as that of gravity. If we imagine that it is an inverse square law force, compressing matter, whether in a neutron star or at the RHIC, would be able to reproduce the non-viscous behavior that we see with helium superfluids. The search for a fifth force that opposes gravity with roughly the same magnitude is ongoing with no clear result so far (http://en.wikipedia.org/wiki/Fifth_force
). I don't know of any attempt to incorporate such a fifth force into our understanding of GR at this time. So the answer to your first question is that there are some who would very much like to see a fifth force, but results thus far are inconclusive due to the weakness that such a fifth force is expected to have at normal pressures and densities.
The second question has to do with theories, but I would like to consider only GR. The discovery of dark energy, though no such potential force had been identified during Einstein's life, did not upset GR because its behavior was consistent with the cosmological constant term. GR is well-defined only up to the Schwarzschild radius. A force which prevented a singularity (not the singularity at 2M which depends on the coordinate system which you have pointed out, but the singularity at 0) would not necessarily do any harm to GR just as dark energy has done no harm to GR. The question would be whether or not such a force would have an effect at lower matter densities where we do depend on GR, outside the Schwarzschild radius. Extremely careful measurements at the University of Washington (http://www.npl.washington.edu/eotwash/
) so far indicate no additional forces at normal temperatures and densities.
I am suggesting the existence of a fifth force (sixth if we count dark energy). The possibility of a fifth force is not new. If I am suggesting anything new, it is that this fifth force will only be seen at extreme density or at very low temperature. Thanks for your response, by the way. I enjoyed thinking about the questions that you posed.