I don't think there is a danger of Livingstone running as an independent should he lose the selection. Nothing is ever certain with him but in 2000 he was prevented from standing as Labour's candidate. Losing a fair and open selection would be different. It would also be wrong of Livingstone to hold the threat of running as an independent over the Labour Party to force a result in the selection. There is no evidence this is the case, though scarred by the split in 2000, many in the London Labour Party fear a repeat.
King's appeal, as detailed by Dave Hill, is toward the younger members in the Labour Party. It is unclear as yet whether this enthusiasm is shared by Londoners in general and, importantly, those living in outer London boroughs.
There isn't too long to go and this race will run parallel to that for Labour leader, though the winner is (perhaps) likely to taste power sooner than the victor of the leadership election if the hype of the Con-Dem coalition can carry it through for a full five years.