Thanks for the reply Felix! So I got my DPST switch and it ALMOST works. When the battery is the mode where it'll give power instantly, everything is great. However, when the batter goes into "sleep" mode, something strange happens:
Closing the switch wakes up the battery and the ATXRaspi. However, the ATXRaspi only provides power to the Pi for 6 seconds, then shuts off! Every time I press the button this happens. And during that 6 second window, if I press the button again, I get another 6 seconds, until I stop pressing the button and then it shuts off again (6 seconds after the last press). If I pull power from the ATXRaspi and plug it back in, it works as expected (press to start, long press to shutdown, etc).
I made a video to show it happening:
I also tried (but don't show in the video) unplugging the ATXRaspi from the battery and then holding the power button down while plugging in the ATXRaspiâthe same thing happens (power for 6 seconds only)! It's also stuck in the same mode where now every time the switch is closed it only powers for 6 seconds. I have to unplug the ATXRaspi to "reset" it. It seems like your description below of the ATXRaspi only needing a HIGH should account for this scenario, but something isn't right...
Maybe there's a simple circuit I can put between the switch and the ATXRaspi to delay that start signal by a half a second or something?