I don't know any autothrottle system that will not go to idle if you command a slower speed. The Boeing operates the same as the Airbus in that respect. Also, if you turn the autothrust off on the Airbus you can see engine response. If I have a suspected engine problem on both engines I sure a heck am going to move the thrust levers around to see if I get something. I would definitely go to TOGA.
Just remember, Sully was flying the jet, talking on the radio, turning on the APU. The F/O had 35 hours on type and was fumbling with the QRH.