Got similar problem. battery voltage measured directly is ~13.8V (engine on), regardless off batt. load (lights, defrost etc). But on dash (in test mode) batt. volatge is lower ~13.6V and now the funny part - when i turn on headlight it drops to 12.9-13.0V and engine stalls. I turned on all electrical systems (but no lights) and it was still 13.6V on my dash.
Measuring volts on battery it was at all time 13.8V (headlights on, etc).
Looks like ECU connection problem, but why does it occure just with the headlights on? Current flow is also OK 4.2Amp = ~55W per one bulb.

mondeo 2002 mk3 2.0 petrol