Neurologists (to my surprise) are hit and miss on properly diagnosing CRPS.
If you do go to one, do your research and go to one that specializes and has experience in CRPS. Many neurologists have sub-specialties that they practice within.
You could also try a different Pain Management doctor.
The ice can make it feel better (temporarily), but it really is not a good idea. That has been pretty firmly established. Temporary use of ice is one of the treatments of choice for ACUTE, but not CHRONIC pain.
Have you seen Hooshmand's site yet? Some pretty good information on there.
http://rsdrx.com/CRPSABSTRACT.htm