Been there, done that. As with anything that concerns my ex with my daughter, I look at it as doing it for my daughter. I love making her happy and her being able to do something for him makes her happy. Yes, he is a jerk, this is why he is my ex but he is still her father that she loves dearly and I love her so I do things for her, not him.
I go as far as purchasing holiday gifts for their whole family with her name as the from. She has been through enough with our divorce and needs to see that no matter what our differences, people can still get along. Perhaps if the world would do this, then there were be no wars.
In the end, you have to do whatever you are comfortable with but this is what I would have and have done.