According to the AZ G&F regulations you can take big game in AZ with "centerfire rifle". (AZ 17-301 through AZ 17-309) Pert this definition, it literally says you can use a .17 Remington if you want to. Personally, anybody that would hunt big game with anything less than .243 caliber is on the ragged edge of hunter ethics. JMHO, of course. An ethical hunter will always use a caliber sufficient for a clean kill on the animal he is hunting.
A long time AZ resident, I too, have hunted Coues Deer in years past. It is few Coues Deer that will tip the scale at 100 lbs. on the hoof and far fewer that will be near 100 lbs after field dressing. I have always found the .243 Win. to be an excellent round for these little guys. A 95 grain bullet is plenty of mass and the .243 is a very accurate, very flat shooter to 250 yards.
Most Coues Deer will be seen at less than 150 yards, however, if you have an area that you can sit and glass with an excellent field of view, it is possible you might see one out to 400 yards. The hardest part with these tiny deer is being able to see the antlers at any distance.
On a side note, I once helped a friend butcher a nice 6x6 bull elk. We found a flattened and partially embedded .22 caliber bullet on one of the neck vertebrae from a previous season. Makes me wonder what kind of pain and for how long the animal suffered.
Poper
