I was just thinking about this particular question, and I think I came up with an answer you probably won't like: it doesn't matter. Unfortunately for us (or fortunately?), anything that happens inside an event horizon is completely unknowable to the outside. While we can have a picture of the metric, it's never observable. Further, the fact that there is a singularity at the center of the black hole in GR indicates not that there is a region of infinite density but that GR as a whole breaks down on that scale.

Also, note that the gravitational waves emitted in the recent merger that we detected are not emitted from inside the event horizon, but from outside of the horizon - otherwise, they wouldn't escape, as you mentioned.

However, one paper I read described the shape of the photon sphere of the merger of (maximally charged) black holes, which has an analytic solution. I would expect (but I haven't done the calculation, so I'm not quite certain) that the event horizon takes on a form similar to that of the photon sphere (but smaller) during the actual merger event. Once the event horizon becomes that of a sphere, what happens on the interior is no longer relevant - everything outside the event horizon cannot tell the difference between a black hole with two singularities or a black hole with one singularity, so for all intents and purposes, we can consider it as having a single singularity.

One question that is possibly worth investigating, however, is exactly how the event horizons combine and the dynamics there - what happens when they overlap, or can they even overlap? Further, an interesting continuation of your question is, if you have two black holes (of equal mass) about to collide with impact parameter zero, and place a point particle exactly between the two bodies, what happens to that point particle? Clearly, the black holes do collide - however, from a distant observer's perspective, the particle can never pass the event horizon of either black hole. So does it get squeezed out? Spread out equally over both horizons? That's more to consider.

I was just thinking about this particular question, and I think I came up with an answer you probably won't like: it doesn't matter. Unfortunately for us (or fortunately?), anything that happens inside an event horizon is completely unknowable to the outside. While we can have a picture of the metric inside the event horizon for some distance, we should expect that the classical approximation breaks down the further we get. The fact that there is a singularity at the center of the black hole in GR indicates not that there is a region of infinite density but that GR as a whole breaks down on that scale.

Also, note that the gravitational waves emitted in the recent merger that we detected are not emitted from inside the event horizon, but from outside of the horizon - otherwise, they wouldn't escape, as you mentioned.

However, one paper I read described the shape of the photon sphere of the merger of (maximally charged) black holes, which has an analytic solution. I would expect (but I haven't done the calculation, so I'm not quite certain) that the event horizon takes on a form similar to that of the photon sphere (but smaller) during the actual merger event. Once the event horizon becomes that of a sphere, what happens on the interior is no longer relevant - everything outside the event horizon cannot tell the difference between a black hole with two singularities or a black hole with one singularity, so for all intents and purposes, we can consider it as having a single singularity.

One question that is possibly worth investigating, however, is exactly how the event horizons combine and the dynamics there - what happens when they overlap, or can they even overlap? Further, an interesting continuation of your question is, if you have two black holes (of equal mass) about to collide with impact parameter zero, and place a point particle exactly between the two bodies, what happens to that point particle? Clearly, the black holes do collide - however, from a distant observer's perspective, the particle can never pass the event horizon of either black hole. So does it get squeezed out? Spread out equally over both horizons? That's more to consider.