Can I drive nitinol wire with an LED driver?
I'd suggest LED CC drivers would be unsuitable for your application since your voltage requirement is so low.
You could use a simple PWM solution but this is completely unacceptable due to the high pulse current. (0.83 Ohms will result in 14 Amp pulses, which exceeds your wire current capability (may act like a fuse) so you need to have a switching solution with a step down inductor which increases complexity)  

Looking at the Nitinol datasheet, you should have a maximum operating current of 1 Amp for your wire.
With your off resistance of 0.83 Ohms that requires a voltage of about 830 mV.

Note: The datasheet says your wire is 0.47 Ohms per inch ...so I'm not sure your question has the right values. Why is the resistance not 1.42 Ohms?? 



You could design a bespoke constant current switching regulator for the purpose, but I'd suggest a quite reasonable solution would be to use a TI LM2596 based Buck regulator which supplies a minimum voltage output of 1.235 V. 
You could simply use a 0.5 Ohm (1 W) series resistor so you pass 1 Amp when the Nitinol wire is cold. It's resistance goes down as it heats up so you can adjust the series resistor to ensure you don't overcurrent the wire when hot.

There are many LM2596 based Buck regulators on Ebay (I use them all the time as replacements for LM78XX regulators) at very cheap prices (around $1.50). There are also plenty of constant voltage/current modules, but they are really based on a foldback current design which I doubt you actually need.
You could design you own of course ...but I'd suggest the constant voltage modules would be fine for the task. 

Be aware that no matter what approach you take, there will be pulse currents of at least 1 Amp drawn from your power supply. You may need to ensure you have sufficiently large output capacitors on your 12 V supply since it is limited to 2A.