Very, very fiddley. I've learned to hate flat flex connectors. That said, I've also got very, very good at connecting them after assembling touchscreens for months.
We've been investigating a volume limit on the Picade PCB since at max volume, with a high input volume it will pull something ridiculous like 1.3A. It doesn't take a mathematician to figure that 1.3A + 700mA is already 2A, without even factoring in the screen. D'oh!
What's happening is that the Picade PCB is peaking over 600mA and tripping your Pi's USB current limiter, which defaults to 600mA but can be raised to 1.2A with (and I'm not sure which works) the following line in /boot/config.txt:
safe_mode_gpio=4
or
max_usb_current=1
Raising to 1.2A may solve your problem, or if your output volume is particularly loud it may just just the amp a higher limit to hit.
There are a dozen other ways to power the whole setup which may help, too, and we're looking into what might work best. Simple things like running the Picade PCB and screen through a good powered USB hub will make a world of difference.