Did Misato end up using the DSS Choker after all?
Yes: Misato activates the choker when the Wunder is unable to pursue Eva-13 due to engine trouble, leaving her out of options. The choker is also set to Misato's personal passcode.
No: The choker automatically detects Awakenings and triggered itself, tying with the "Awakening Analysis Mode" function. It took so long to actually kill Kaworu because he used his powers to stall the detonation as long as possible. Furthermore, the laws of cinema dictate that any use of the device by Misato would have included an establishing shot -- which is notably absent.
Obviously, if Misato did end up pulling the Trigger she would know that it wasn't Shinji since the DSS choker detected a Blue pattern (though it was somewhat ambiguous to if they knew Shinji wasn't an Angelic abomination) which definitely checks out since Kaworu is definitely Angelic.
Assuming Misato DID pull the DSS choker and it wasn't automatically designed to go off?
1) Assuming she knew it wasn't Shinji, would he blame her for killing (Supposedly) the last person who ever cared for him? Denial and deflection is a step on the way to acceptance.
2) Assuming she was out of options and pulled the trigger, assuming she still thought it was Shinji wearing it when she pulled the trigger. How will she react to finding out he's alive? How will Shinji react to finding out she pulled the trigger believing she was killing him and not Kaworu?