If a Skyseeker designates his double weapon as his Ancestral Weapon, would he get the extra damage with both ends, or only with one?
Too bad you never got any answers for this almost a year ago, since it actually is quite relevant question.
Anyway, the way I read it, most similar class features state explicitly if they affected only one end of a double weapon. By "the rule of omission" it quite clearly says that your ancestral weapon would deal extra damage with each hit, whether or not the hits came from a single-head weapon or either end of a double weapon.
So yes, Skyseeker designates one weapon as his Ancestral Weapon, and a double-weapon is still one weapon in itself, even though it has two ends.
PS. sorry for being a necromancer!