Just found this: https://www.raspberrypi.org/forums/viewtopic.php?t=50470
Kinda very relevant, but no mention of the Pi restarting at the instant the main and backup supplies switch over. Maybe earlier versions of the Pi don't have brownout resets?
Oh, and then there's this: https://thepihut.com/products/ups-pico?variant=20063191203902
...which I could just buy. :(
The RPi4 seems to use the MxL7704 PMIC, instead of the APX803 supervisory IC used in earlier versions. The 7704 does not have a brown-out reset line, so for my Pi to be resetting itself, something must be triggering that reset indirectly by monitoring the PGOOD lines. It's probably not helped by the supply already being low enough for the lightning bolt to be showing (<4.65V), likely due to the drop across the Schottky. I'll have to get the thing hooked up to a 'scope to see how big the glitch is. Adding 1000uF to the 5V rail did not cure it.