No, as the More Points bonus terms require people to choose Hyatt points to get the More Points bonus. But you can get the More Points and the 1,000 AA miles, as that bonus doesn’t require one to choose miles.
In other words, if you choose Hyatt points, you get the base points, the More Points bonus, and the 1,000 AA miles bonus. If you choose miles, you get the 500 miles plus the 1,000 AA miles.