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So long as the archetype does not alter Rage, and only replaces class features that the Unchained version possesses, it's allowed.
Invulnerable Rager does not alter Rage, so you're good there. It does replace Trap Sense, which the Unchained Barbarian does not have.
Luckily, you're still good: "[Danger Sense] counts as trap sense for any feat or class prerequisite, and can be replaced by any archetype class feature that replaces trap sense."
Here's the Society Unchained Blog for further reading.