I thought I'd give an update.
I added four .470 capacitors in paralel, but the gun was still turning itself off. Then I added the diode, and.. it worked! The gun remains stable.
...at least.. while it was on a beefy lab power supply. It indicates that when firing the gun uses about .6 amps, and it prefers to be around 8.5 volts (6 AA batteries). 9 volts seems to be too much.
I tried to switch it to the intended portable design, with a USB PD decoy feeding into a 4A up-down adjustable buck converter.
But then it crashes again.
For example, I tried this:
55W steam deck USB-C PD power supply ->
adjustable decoy set to 15 or 20 volt.
down-converting that to 8.5v with the buck converter.
Maybe all this power converting means that it can't provide the sudden increases in power when the motor spins up?
Perhaps using a linear DC-DC converter work better? Or maybe the USB-PD decoy just can't handle it? It's a mystery.