Delivery can make the difference between affectionate parody and harsh parody.
Most of Weird Al's songs don't really make fun of the original performer. Most of 'em are really about food, or surgeons, or the Amish, or the Beverley Hillbillies. Only a couple of 'em actually target the perceived flaws of the actual singer (e.g. his version of "Smells Like Teen Spirit" which makes fun of Kurt Cobain's somewhat incomprehensible drawl).
This song did strike me as a little more critical than most of what I remember (though Smells Like Nirvana did come to mind), but hey, that's obvious from the lyrics. I wouldn't fault her for saying no at that point; instead she refused to respond until he'd done the recording.
So I agree with Al, it was a stupid demand to make (especially considering that she should know that recording a song isn't cheap), and Al's reaction isn't really about the money; it's about punishing her for either thoughtlessness or meanness, and also a bit about getting a bit of press for himself.
I suppose I'm not terribly impressed with either of them.
On what grounds would she reject the song, but be ok with the lyrics? Because by asking him to sing it she's basically said "I'm ok with the words. But I might not be ok with the delivery."