Twelve year olds (especially girls) have a very finely tuned sense of justice and fairness.
In her eyes, she was only doing a nice thing in playing with her cousin and talking to her affectionately and then she gets snapped at for her efforts. She'll be doubly chagrined by what she perceives as the double standard of her uncle being allowed to shorten the name when she is not.
Sometimes adults do speak more sharply to children than they would to other adults in the same situation, and I think that children become very sensitive to this.
The thing is, it's not an equal relationship. If your sister in law had snapped, "It's Isobel, not Izzy!" at you, you would probably have said gently, "Ok, sorry. No need to snap!"
But your daughter does not have the status to do this. Responding in any way other than a very contrite "sorry" would be seen as cheeky and rude.
I suspect your daughter knows this. It's one of the great injustices of the world (when you're twelve) that adults are not always perfect and sometimes say or do the wrong things, but you are not allowed to question them or argue with them about it.
So...yes, your daughter is probably being a bit of a drama queen. Who wasn't at that age? Her upset is disproportionate to the incident. But it doesn't mean that your sister in law didn't snap at her, or that she did so unfairly.
I would say that a gentle, "Aunty X shouldn't have snapped at you love. I know you were only trying to be nice. She's probably really tired because looking after a baby is hard sometimes, so don't take it to heart. Just remember to call the baby Isobel in future!" is the best approach. It means that her upset is acknowledged without allowing the situation to be blown out of all proportion.