Simple answer is no. It does not magically transmute things that specifically ask for Con into Chr.
"No Constitution score. Undead use their Charisma score in place of their Constitution score when calculating hit points, Fortitude saves, and any special ability that relies on Constitution (such as when calculating a breath weapon's DC)."
It is very quite specific, and not "charisma modifier in place of all con modifiers" as you put it, which is quite a bit broader.
What more barbarians make horrible Undead. "Immunity to all mind-affecting effects (charms, compulsions, morale effects, patterns, and phantasms)." Your rage is a morale effect.
Though its not like the concept is unheard of. There exists a random vampire NPC that gets almost exactly what you were asking for, but It was basically made custom and there is no method of obtaining its abilities.