Engagement party or not, a bit late home from work or not she does not have absolute control of the child as it is his child as well.
It seems to me that she is behaving in a very childish and petulant manner. The father is, from the details, not an absent father and is involved in the childs life.
As would be expected he was celebrating a happy occation and wanted his child there. So he got a bit carried away and time slipped by! So bloody what? The child was probably having a good time as well!!
This mother is setting a bad example to her child. Not only is she showing her nasty side (perhaps it is her natural default.... who knows) she could possibly be turning her child into a hateful and spiteful person.
Like so any people that have split she appears to be rather bitter about it and is punishing him by being somewhat unreasonable.
If, for instance, they had remained together and she was away visiting friends and he had to pick the child up from school and he was running late it would be reasonable that he ask a family member to pick them up. There would not be a problem. Now, suddenly because they are no longer together she has decided that isn't good enough and he has to be absolutly accountable to her demands and nothihg else matters.
Nasty, bitter, petulant and downright childish behavour from the mother.
If I were him I would go for shared custody.