Anyone else seen this one before, or have any tips for how to diagnose the problem?
When I start the car (2004 L322), the touch screen head unit starts up, works just fine for around 10 seconds, then shuts down as though it has no power.

Check all the connectors in the rear left cubby in the boot -- I really mean all of them, it's probably best to take all modules out and even inspect inside the units too Current: 2011 FFRR Vogue SE 4.4 TDV8
Previous: -
2004 FFRR Vogue 4.4 V8 with LPG
1997 P38 4.0 SE with LPG
1997 P38 4.6 Vogue with LPG

What’s the bare minimum modules that need to be connected for the head unit to operate?

Just the BM54 if I remember correctly -- but best to have that out too to check Current: 2011 FFRR Vogue SE 4.4 TDV8
Previous: -
2004 FFRR Vogue 4.4 V8 with LPG
1997 P38 4.0 SE with LPG
1997 P38 4.6 Vogue with LPG

Just to close this up. I disconnected the barrel for a while, then plugged it back in, and the unit started functioning fine again.