Can Eleanor use UC to use her ability twice in immediate succession? I am thinking not, because there is no action window to use UC while revealing cards, so if a second treachery card is revealed, Eleanor is exhausted and can't trigger her ability.

Correct. Per the timing chart at the end of the rule book, there's no action window during staging (i.e. the step entitled "Encounter deck reveals one card per player"). You can trigger Eleanor's ability because it is a response, but you won't be able to use Unexpected Courage to ready her during that step.