Hi and Thank you all! OP here - sorry for the delay in responding!
The kiddo has now started at the school.

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We do have an option to fly her on Southwest, but for the dates we need AA prices out better than Southwest. Also - I'm trying to use up my BA Avios for some AA redemptions. So, many of these AA flights will be award tickets and not revenue (so no hopes of getting status!).
Anyways, yes, so I've added my child as an AU on my Amex Plat - to get her credit history built up. Will give it a few months and try to apply for Citi or Barclays card. Even a $1000 credit limit is fine (or preferred actually ;-) ).
Thanks all for chiming in!
Best Wishes