Binary black hole merger viewed from inside the event horizon
At the event horizon the "coordinate" speed of light according to distant observers is zero. So if you were viewing the merger from inside the event horizon, and I was somehow viewing you from where I'm sitting via my bubble of artistic licence, I would say you have no view at all. There's a conflict between this and "the proper speed is c". See Kevin Brown's The Formation and Growth of Black Holes and note the mention of future infinity. As far as I can tell it takes you forever to see anything. You haven't seen it yet, and you never ever will.
How did the metric evolve inside the event horizons of the black holes whose merger caused the GW150914 signal?
I don't know. But the black hole is a black hole for a reason, and it isn't because space is falling down. It's because the "coordinate" speed of light is zero. Which suggests there's no evolution inside the event horizon.
In principle the Schwarzschild metric of a non-rotating black hole is known inside the event horizon
I rather thought that the Schwarzschild metric was only valid outside the event horizon, as per Graham Reid's comment.
although the analogous Kerr solution for rotating black holes seems to have unphysical properties in this region.
Agreed. Take a look at the Einstein digital papers, and you can read this from 1920: "the curvature of light rays occurs only in spaces where the speed of light is spatially variable". He didn't use the word coordinate, which suggests that at the event horizon, the speed of light is zero. Which suggests that the spin rate is zero too.
Is it possible to at least simulate the dynamics of the metric inside the event horizon during a black hole merger, and get a meaningful answer?
Maybe. But my reading of the Einstein digital papers is that there are no dynamics. Rather counterintuitively, the descending photon slows down. See this PhysicsFAQ article by editor Don Koks.
If so, what happens and what would an observer inside the event horizon see?
Maybe there's a way we can have our cake and eat it, in that the expansion of the universe is not limited to the speed of light. But I'm still not hopeful about that observer seeing anything any time soon.
If not, why not?
Because the light doesn't get out. Because gravitational time dilation is infinite. Because the coordinate speed of light is zero.
The main inspiration for the question is my semi-Newtonian intuition that once the event horizons merge, the two singularities would rapidly orbit each other inside the event horizon, and eventually crash into each other due to emission of gravitational waves (which of course must remain trapped inside the event horizon). I highly doubt that this intuition is correct. Can general relativity give us a better answer?
I think it can, but I side with the frozen-star interpretation, and I'm currently in the minority. Again see Kevin Brown's The Formation and Growth of Black Holes: "Incidentally, we should perhaps qualify our dismissal of the "frozen star" interpretation, because it does (arguably) give a serviceable account of phenomena outside the event horizon, at least for an eternal static configuration. Historically the two most common conceptual models for general relativity have been the "geometric interpretation" (as originally conceived by Einstein) and the "field interpretation" (patterned after the quantum field theories of the other fundamental interactions). These two views are operationally equivalent outside event horizons, but they tend to lead to different conceptions of the limit of gravitational collapse. According to the field interpretation, a clock runs increasingly slowly as it approaches the event horizon (due to the strength of the field), and the natural "limit" of this process is that the clock asymptotically approaches "full stop" (i.e., running at a rate of zero). It continues to exist for the rest of time, but it's 'frozen'..."
What this frozen-star interpretation says is that there aren't any point-singularities. Kevin Brown suggests Einstein would have sided with the other more common interpretation, but I don't think he would. When you drop your pencil, it falls down because the speed of light at the floor is less than the speed of light in front of your face. But when you're at the event horizon, the speed of light in front of your face is zero. And it can't go lower than that.