You taken a lovely invite and turned it into a confrontation. You imply your DIL is trouble making [actually she does have a say in her child's birthday arrangements too, it's her life too] and you will be 'tense' on Saturday. That is not being cooperative or pleasant at all. You're going to create an atmosphere from the off. Don't be a nightmare MIL.
This is all because your son wishes that your grandchild has an outing with it's parents and grandparents! Your daughter seems to be making a meal of this and winding you up. Don't get involved. it's not about you, or her, it's about your son wanting to have your company...and you are rejecting him. There is a danger you will see much less of your grandchild in the future if you keep rubbishing your son's life, needs and views on how he wishes to lead his life.
So son and daughter don't get on, it happens all the time. let them negotiate it as they wish...not how you want. Daughter sounds like she is over reacting anyway, too much drama.
My 2 sons aren't close, and don't see each other much - rarely on family occasions. But there's a respect for their differences. I don't take sides, my relationship with both is good and it's their relationship between them to address, not mine.
Seems you are giving yourself a huge dose of stress by wanting so much control. It's making you unwell, let go.