Thanks for the tips (and the driving tips at Buttonwillow). I have aftermarket poly transmission mounts with aluminum enforcer cups, but they are not bolt-through mounts and not the stiffest poly option. My motor mounts were stock Lemforder with ~18,000 miles on them, but after this happened on Saturday, on Sunday I switched to BimmerWorld solid rubber (but not bolt-through) motor mounts, since I had a set laying around.
Reviewing my video, the problems going to 2nd happened only after my big tank slapper. But then again it seems I could downshift right as I started turn-in but not during the middle of the skidpad turn. So maybe one of my transmission mounts is broken now, or maybe I just shouldn't try to downshif mid-turn. I looked at the transmission mounts Sunday but didn't take them out. I should go back under the car and remove the transmission support brace to verify those mounts are okay.
Lowering my car's resale value one track day at a time...