@Anonymous, as others have mentioned, Chase will likely be happy to have you once your AMEX card becomes a year old. They're going to disregard the AU account. The other thing they like is three to six months of gardening (not applying for or obtaining new credit) before applying for one of their cards.
To get a CSP and its minimum $5,000 limit, they like to see "experience" with a similar limit on at least one other card. There's a good chance that if your AMEX isn't at $5,000 right now, it can be by the time it's a year old. Read the first few pages and the last few pages of The Definitive Amex 3X CLI Guide for details.
As you suggest, a lower tier Chase card may be the ticket to a higher tier card. But for lower tier cards, they still like to see the year of history on a card where you're the primary user. However, if you already have a Chase card in tow, they may feel inclined to grant the $5,000 limit even though your other cards havn't yet reached that point.
Getting another card before applying for Chase is a good idea. I think a good time to do that is anytime after your AMEX card shows up on your reports. That'll likely happen after your second statement. Getting a second card anytime within the next three to six months will give you more than enough gardening time for Chase.
Because you're seeing a FICO from AMEX, you'd now have FICO scores from all three bureaus. You can get your three FICO8s along with the three corresponding reports by signing up for a $1 trial at CreditCheck Total, then canceling within the trial period.
Having said that, I don't think your scores are going to produce any unpleasant surprises. As long as your payment history is perfect and your utilization is in check, the important thing is going to be the age of your profile.
