For 1mS at 350mA, you're likely to need a large capacitor to supply the current.
9V batteries have a relatively high internal impedance, and it's likely that the voltage during the current pulse is sagging rather a lot.
For best results while experimenting, a bench power supply capable of 1-2A would be good, a second battery for the flipdot would work as well. You might use AA cells in a holder for that one, and then the voltage sags would be smaller, and not cause processor resets or other weird behavior.