I decided to see if I could "cheat" by make an electronic automatic trigger of some sort. I took the game box apart, and using a multimeter, quickly determined that when you press a button, it just connects one point in the circuit to the ground. So make an electronic automatic trigger should be pretty easy.
I drilled a hole on the side on the game box, and run 4 wires in, soldering them to the power, ground, A button contact, and B button contact (unused in my circuit) of the game circuit. I wanted to build a simple oscillator circuit using the 555 IC, but I was out of 555 so I used one half of a 556 (which contains two 555's) instead. The "trigger" circuit is your basic 555 astable mode circuit:
When the output is high, it turns on the transistor and connects the button contact to the ground, thus electrically "press" the button. The rate of fire, using the component values shown, is about 18hz, or about 1000 rounds per minute. That's more than enough for the games since they allow far fewer bullets on screen at a time.
I had to solder the power wire to the power pin of the 556 chip, but everything else was jump-wired on a small breadboard duct-taped to the game box:
