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Being added on her card would help you if all of the following are true:
(1) Her card is much older than your oldest account. (E.g. if her card is ten years old and your oldest account is two years old.)
(2) She has never had a late payment on that card.
(3) You are confident that she would never have a late payment in the future.
(4) Her current balance on that card is low (definitely < 30%, far better would be < 9%)
(5) You are confident that she could keep her balance low in the future.
You are right that being added as an AU to a nearly maxed card would hurt you.
Your idea about her getting the Slate is good, but only if she has a disciplined plan in place for then steadily paying the Slate off while also steadily reducing her CC debt on all her other cards. Otherwise the BT to the Slate will just represent a new source of credit so that she can buy more things, and she will ultimately be in worse shape than before.