I don't know if I've found one theory that is completely satisfying, but the best one so far would seem to be the idea (mentioned above) that Shambhala is a way to connect all humans to the nagari network, perhaps forming some sort of unified psyche. Assuming that humans in the GTVA become connected as well, this could very well be a way to end the war without having to fight it out, as the Elders would seem to prefer.
This idea seems to be in line with a lot of the fiction. For example, as per AoA, the vishnans seem to believe that humans have the potential to reach a state of enlightenment, and by doing so can take the place of the brahmans in some sort of heavenly trinity between the shivans, vishnanas, and humans. It's indicated that the vishnans, through nagari contact with the Elders, are cultivating humanity towards this state of enlightenment. It is therefore plausible that this level of nagari-connectedness between humans is a major milestone on this path to enlightenment. This would somewhat explain why Shambhala wasn't making much progress until the highly nagari-sensitive Samuel Bei showed up with the 14th battlegroup (although it doesn't explain why nagari sensitivity amongst the first alpha one, the vasudan pilots, and the Elders wasn't adequate for this progress).
Obviously Ken has other designs. His deal with the shivans is based on having humanity do the "one thing the shivans can't". Considering the "War in Heaven" that is currently going on between the shivans and vishnans in this story, my best guess is that this task involves injury/desctruction to the vishnans by humans. How could humans, or Laporte in particular, damage/destroy the vishnans? I'm not sure. We know that the vishnans inhabit and control the boundaries between universes, and perhaps subspace as well. We also know that they can communicate through nagari contact.
Perhaps Laporte has been meticulously cultivated by Ken to use her off-the-chart nagari abilities to either "hack" the vishnans, or somehow mess with the boundary world that they inhabit. Perhaps she needs to assemble a large battlegroup (as Ken tells her in Universal Truth) to punch through GTVA space and reach some location where this can be accomplished? Perhaps this location involves that newly discovered ancient Knossos portal in Ross 128 that was mentioned in AoA?
One problem with this theory is that the vasudans play almost no role here. It is mentioned somewhere (a tech entry?) that one of the vasudan Lucifer pilots was invited to visit Emperor Khonsu II. Between this and all the tech info about vasudan mysticism and the Hammer of Light, it seems likely that vasudans will end up playing a huge role in this as well. As a matter of fact, vasudans seem to be more enlightened than humans. So maybe I've got this all backwards, and it's really the vasudans who are supposed to take the place of the brahmans.
One more quick thought: At the end of WiH R1, the two Beis mention that the capture of the Anemoi logistics ship was a big help for Project Shambhala. The Federation also seems to be holding back a huge chunk of the military to aid in its completion. This leads me to believe that, whether weapon or not, whatever they are building must be *very large* (analogous to the Crucible in Mass Effect 3). Why else would they need all those assets?