The task group will display the "out of fuel" message if any ship in it is out of fuel, so that is working as designed.
Have you checked the box in ship design that says "Tanker" for your tanker design?
If yes then you should be able to say "refuel from own tankers" in the orders. Also the conditional orders to refuel from the nearest tanker should use its own, though I'm not 100% sure of this.
Another option is to detach the tanker and give it orders to refuel the fleet then join.
But there is otherwise no auto refuel function (equilize fuel does it "sort of"). I would say your best bet is to use "move to jump point x", "refuel from own tankers", then continue on. This way it will have its tanks topped up every jump.
But one standard fuel tank of 50,000 l for a 12,000 ton carrier and its fighters is really far far too little. My 9000 ton destroyers have 600,000 l of fuel. Also what happens if someone or something kills your tanker? That will mission kill your fleet right there.