Myself and my ex weren't married when the kids arrived and I was advised to double barrel their names incase we ended up splitting up. The inevitable happened, we split up and we just dropped his surname for the school, doctors etc. The only legal document they have with his name is their passports and it really wasn't worth all the hassle to have his name removed..
BTW wasn't my idea to remove his name, its what my children wanted. They have nothing to do with him for years now and don't want a name associated with him