I think this father is an being an arse, he's dragging his child through the media, and this problem is of his own making.
They accepted the invitation and then took up a better offer.
He realised he had made arrangements with the grandparents before the day of the party, and still made no effort to contact the party family to cancel while they had time to offer the place to somebody else.
They say they put all their details on the invitation, so he's clearly lost it or thrown it away.
But he could still have called the venue and asked them to pass a message on, contacted the family via Facebook (as he claims his partner is doing now), or asked around the other parents to see if they had a contact number.
And it's not like they forgot the party or had an illness or emergency to deal with.
He just asked his son on the day if he wanted to go or not, and let the other family down without a second thought.
If that were us I would think that he was rude and his child was spoilt, and that all of them were selfish.
I wouldn't invoice them, but I wouldn't be in any sort of hurry to invite them to anything else in the future.
He's created this issue, by being selfish enough to let someone spend a lot of money on his child and then drop out without warning or apology. It takes time and money to plan a party like this, and with limited spaces every no-show is a let down for the child whose party it is.
And the other mother has had the balls to call him out on his behaviour in quite a direct way.
I don't believe she expects him to pay the bill, but I do think she used it to prove the point that he had cost her a lot of money and let her child down.
And now he's driving around at work telling anybody who will listen that she's done this and gone running to the press and the media.
He's an arse.