The 1A measurement was made using a multimeter at the 9V output when it is not connected to the Arduino
Nonono! When the multimeter is in amp-mode it behaves like a straight wire. If you connect the multimeter terminals to the power supply + and - like you would when measuring voltage, you're basically making a short circuit! Luckily the 9V regulator can't give more than 1A and the multimeter could cope with it. Had you made this "measurement" on the battery you would've damaged the multimeter or at least burnt its protection fuse!
What you need to do is connect the multimeter like it was a simple wire going from the 9V regulator to the Arduino. For example, multimeter + to supply +, multimeter - to Arduino +.