Hello, I hope the following is of assistance.
The ABS and TCS lights will be on due to low battery voltage as there is a fault with the charging system. The Duratec V6 from 2003 onwards has an inline fuse in the main lead between the starter motor and alternator, check the lead using volt drop method as a continuity check usually doesn't show a fault with a fuse of this size (175A) there should be 12 volts at the main B+ terminal of the alternator, there should also be 12 volts at pin 3 of the small connector at the alternator, this is supplied by fuse 30 in the fuse box under the bonnet.
If there isn't 12 volts at the main B+ terminal and the volt drop reading is more than 0.5 volts then check at the starter motor for 12 volts (I suspect this will be OK) if there is then the inline fuse has blown it will be part of the lead between the alternator and starter.
From your description I suspect this lead/fuse will be the problem as any concern with the smart charge system still allows the alternator to charge at a steady 13.5 volts which means the car would continue to run instead of flattening the battery.
Also confirm the charge rate from the back of the alternator and at the battery.
I hope this helps but if you have further questions let me know and I'll reply as soon as I'm able to.