The behavior appears to be the same when connected directly to the MOD rather than chaining off the footswitch.
I have also tried completely disconnecting the display, and loading a modified version of the ‘Blink’ sketch to alternate a different pin high and low (the LED on pin 13 appeared to be stuck on when powered via CC), and measured that with my multimeter. When I power the Mega through USB I see the switching from ~0V to ~5.8V, when I power via the CC port I just get a constant ~1.4V (although the 5V locations on the board are showing a good ~5.9V).
So I then switched to an Uno that I had laying around. Running the same program on it (ie Blinking a digital pin) seems to be ok when powered via CC.
So, it seems to be quirk of using the Mega 2560 board and powering it through the CC shield.
Further thoughts much appreciated!