I was thinking about this while I was convincing myself to get up this morning, and I realized it might not be the animations at all.
Last night when I was playing some more, using the new build, too, I made it to the final mission. I restarted that mission once, and the issue set in, but it wasn't just happening with communications animations on screen. I was getting a small jitter just as the beam sounds started.
Now, I have it set to preload mission sounds, so I'm wondering if there's another issue here, something with it not clearing the sounds out of memory before reloading everything? Or perhaps with it not preloading sounds correctly when a mission reloads to play again?