The roundworm (c. elegans) only has 302 neurons, and about 7,000 synapses, but is capable of social behavior, movement, and reproduction. The entire connectome has been mapped, and we understand how many of these behaviors work without having to resort to additional ontological entities like your "third way".
If this complex behavior can be explained using only 302 neurons, I have no doubt that the complexity of human behavior and consciousness can be explained using 100,000,000,000 neurons.
Behavior? Yes. Consciousness? Maybe not. As far as I know, no-one has ever come up with an explanation of how matter can give rise to consciousness. Obviously consciousness is affected by changes in matter — if I take certain drugs, I feel different, etc. — but there is no explanation at all for what mechanisms may give rise to consciousness in the first place, and the very idea of the material giving rise to consciousness might actually not make sense.
This line of thinking begs the question of whether consciousness is some special thing in nature. You only have to look for an explanation if you think there is something to explain.
If I start from the assumption that I am mistaken about what I think consciousness is--that maybe it doesn't exist at all the way I think it does--then I don't have to worry about how matter gives rise to it. I can focus instead on trying to understand where my definition went wrong.
Humans have never lacked for opinions or explanations about what natural things are, or how they got that way. But in the practice of science, these must yield before empirical evidence.
As you note, there is a huge amount of evidence that consciousness is physical. But there is no objective evidence that I experience consciousness as you define it. You just have to take my word for it--and so do I. But maybe I'm wrong.
There is an implicit assumption there that scale is colinear with sophistication. It could be that only 5 neurons give rise to the complexity of consciousness, it could be that the number of neurons is irrelevant but rather their configuration is important. Also, there is a problem there if it explains only some of the behaviors. What happens with the rest, where are they coming from, if we have fully mapped the connectome?
We can't explain everything completely right now. However, for example, we can model behaviors like klinotaxis and movement toward chemical gradients using only 10 neurons [0]! I find it completely fascinating — we don't have the full picture, but we can isolate different behaviors and examine how the neurons contribute to that action.
But it can't all be explained just using neurons. Those neurons exist as part of a body. That body lives in an environment. Those environments include other people. You can't understand the totality human behavior without including bodies and environments. It can't all be reduced to neuroscience.
It's true for the worm as well. You can't understand everything about the worm without emulating the other 600 or so cells, and putting it into environments similar to what biological worms occupy. The lego mindstorm worm is interesting, but it's hardly the natural state of the worm.
If this complex behavior can be explained using only 302 neurons, I have no doubt that the complexity of human behavior and consciousness can be explained using 100,000,000,000 neurons.