Get the Amex accounts paid in full and after some time — perhaps measured in years — apply for Amex cards again.

I know a very high credit score person whose Amex card account was shut down for no apparent reason — other than perhaps there being some very old Amex account default (as in over a decade earlier from the account closures) for the account holder or an AU (perhaps on the AU’s own account) — but a few years later the person and their AUs had multiple, new Amex card accounts. Never did figure out what exactly caused the forced account closure, but it definitely didn’t have to do with chargebacks, bounced checks, or extensive use of the cards’ benefits for purposes beside the loyalty program aspects of the cards.

Having AUs doing way more spend on an account than the primary account holder can be a flag, but that isn’t a show stopper if Amex’s concerns are addressed by an Amex-demanded financial review.