as you described the problem is that the voltage drops quite a few.
it is independent of your hardware.
if your battery won't solve the problem, you should install a thicker grounding wire from battery - to chassis. you might want to try a 1 F capacitor also.
if even this won't work, you surely have transient noise over your power lines.
installing some relevant caps in parallel to all power outputs.
for this all, the easiest method would be to pick an oscilloscope and measure signal shape on power leads over ignition.
...and if you feel this is a problem that anyone should be able to solve in 3 min, I ask you did you ever wonder about why some (if not all) head units/amps/etc DO turn off during ignition...