You don't like her. She knows that. You don't approve of her. She knows that too. 'This woman' is your son's wife and the mother of your grandchild.
This woman, gave birth 3 weeks ago, she is sleep deprived, stressed and nervous as hell. She is going to be left alone, all by herself, with this small baby. She also sounds like she is breast feeding. She is emotionally and physically drained. She is anxious about being on her own. Just like 99.9% of 1st time mom's.
She also has every right to pick her child whenever she feels like. She is not spending 'his' money, she is spending 'their' money, just like she's at home raising 'their' child.
From these few paragraphs it's clear she isn't good enough for your son.
Your DIL obviously knows this, as she's had months if not year's of it. I wouldn't want to spend those first few weeks being a new mum, when I was most vulnerable with someone who doesn't like me. She probley feels the same.