A.V.
I think he's being petty, frankly. HE switched, knowing there was an event for a child (and he didn't get the info before agreeing) and now his wants have come ahead of something important for his kid. I would tell him, "This is our daughter's BEST friend's birthday. Please either drive her to the party or please switch your switched weekend to another weekend so I can take her and she can attend. We can find options that don't hurt our child. Please work with me for her sake."
My SD missed many a party because her mom wouldn't switch and wouldn't take her and it was really sad. Our rule was anything on her mom's time had to be approved by her mom first and SD heard "no" so often that she quit asking. I saw her face when she checked our kitchen calendar to see whose weekend it was. If you and your ex overall get along, please work on this, even if it means that you both now put in more details like location. Remind him that this isn't about you and him. This is about Chickpea. Who is 8. And wants to go to her friend's party. I hope something can be worked out and that a miscommunication doesn't become a power play.
ETA: How far is it, realistically? I ask because in MD, you can drive 10 minutes from some places and be in "another county". Is it feasible for you to take her to the party and then take her to her father's house? Or give him 2 weekends in a row?