A Link to 2003 Neon Schematics can be found on this page:
http://www.techknowman.com/dodge neon/dodge neon.htm
If that's all correct Its showing the 2003 Neon as having its voltage regulator built into the Powertrain Control Module (computer)
Your alternators small green wire with an orange stripe should be reading full battery voltage.
The other small solid green wire gets a variable
ground signal from the computer. This is used to control the alternators output level.
Hey Eric you might just be fighting a voltage drop issue here. An easy test you could run is to compare the voltage exiting the alternators output post (where the big black/grey wire bolts on) to whats actually getting back to the battery.
You'd wanna run this test with the alternator under a heavy load, so turn the headlights on, get the heater fan running on high speed and warm up the engine first so the radiator fan is running. You'll also need the engine at a fast idle (2000rpm)
If your getting more than a 1 volt difference/drop between the Alternator and battery your probably just fighting a poor connection somewhere causing high resistance.
One of the more common causes of this is at the Alternators fusible link. (the Schematic shows this is near the starter or starter relay on your model) If anyone ever tried to jump start the car backwards, It would have burned out this Fuselink. (these fuse links are many times just a short section of a special type wire)
After burning this type of fuselink sometimes shotty mechanics simply use a Butt connector to crimp the fuselink back together causing high resistance problems like this. I've seen cars reading 20 volts at the alternator with only 13 getting back to the battery.
If your alternator is rubber mounted, make sure its ground strap isnt the problem. You can get voltage drop problems from the ground side too...
And make sure that belt isnt just super loose and slipping.