im not saying that this is not possible because dakotas are known for stupid electrical problems, but i still dont see how it can be related. the battery in my truck is complete crap and ive never seen this as being an issue. my truck idles fine. if he keeps on the gas and it runs but dies when he lets off, its a dead give away of a bad IAC. its telling him that its sticking when its cold. which bad parts will do in the cold. replace the IAC and i can almost garantee the problem will dissapear.

or on a side note, try disconnecting the battery for 10-20 min, then reconnect it, and run the engine when its cold. if it runs and idles, its your IAC. another option is intake manifold cleaner. take the IAC out and clean it up with that