But it is happening already, has been for a few months. It isn't being administered by the local councils. Every one of the millions of households with an electricity account is already getting a £66/67 discount off their electricity bill every month. No need to involve council tax or the local authority at all.
The problem is the less than 1 million households that don't have a relationship with an energy company like boaters in a marina, caravaners in a caravan park, people living off the electricity grid etc.