It can be done. I know cause I have one plated in my garage.
It all hinges on this form:
http://www.dmv.ca.gov/forms/reg/reg31.pdf
Reg 31 Verification of Vehicle.
If you go to the DMV to register it as a street legal vehicle, you'll get many forms including Reg 31. Section 6 and 8 are the two most important sections.
6 refers to the FMVSS status (lights, reflectors, horn, etc). This can be worked around with a Brake and Lamp certificate from a licensed dealer.
8 refers to the emissions status and usually the most troublesome since the FE's emission sticker does not certify it for on-road use.
The DMV or CHP can do the inspection. I've heard that DMV's are getting lax about the inspection since they'd rather have your money than be CARB's lackeys. The DMV got a couple hundred bucks out of me that they would not have gotten if they had turned me down.
All that said, you need a Reg 31 that is filled out appropriately to allow the DMV to give you a plate. That's it. Everything else is noise. That form + insurance + reg fees = plate.
The fact that you have an Arizona plate does not simply the process at all. The only thing it may save you is from the brake and lamp check. Maybe
Official DMV page about dual-reg:
http://www.dmv.ca.gov/pubs/reg_hdbk/ch15/ch15_15.htm Scroll down to "On-Highway Registration—The requirements are:"
Good luck.