Late in the game here but in case anyone else is lurking and can offer an idea...
It sounds like diversity across programs is lacking but does anyone have experience with using the peer/diversity mentoring programs at either Goldman or Heinz?
Experience helps but I think the application materials play a much bigger role (i.e., demonstrating leadership capacity in your essays or aligning your goals with offerings in the program). I believe HKS also has some cut offs for the GRE -- they look for 158 at least on the GRE Quant.