We have argued so much about this and he doesn't see it as important to include a link to my family. He says it's a family tradition in his family to have the middle name of your father. But surely he should be accepting of my choices too. He wants me to tell my dad we have changed our mind
So something that one of his ancestors first decided, possibly hundreds of years ago, is of utmost importance, but something that YOU, the child's actual Mother, want, is of no importance? It's not even like it has to be one or the other - it isn't enough for him to get exactly what he wants (which sounds reasonable), he's only happy as long as you specifically DON'T get what YOU want - even though there's no problem whatsoever with both of you having your choice? And it's so important to him that he wants you not to get what you want, when what he wants is already in place and not in any jeopardy, that he has actually been arguing for his right to deny you it?
He shouldn't be 'accepting' of your choices - that makes it sound like he's the one in charge of a 100% joint right and responsibility and you only get any say at all if he deigns to allow it.
He also not only wants YOU to tell your Dad, but to lie and make out that BOTH OF YOU have changed your minds? Having already made your combined decision and informed your respective families, he wants to deliberately snub your Dad by insisting on removing the name that, as PPs have said, will only really appear on official forms and be there as an honour and a gesture, and will not even frequently be used - and he expects YOU to be the one to deliver this snub?
This is very concerning, controlling behaviour. If you don't stand your ground on this one, who knows what else he'll decide he has sole autonomy on and that your opinions and decisions don't matter at all?
If he persists, make him justify his ridiculous behaviour. Tell him that the decision centres around what you each want to name your joint child. You each have one request on the agenda. His request is his name as one middle name, which you have agreed on. Your request is for your father's name as a second middle name, so on what grounds could he possibly object? Force him to tell you straight and justify exactly why an absence of another middle name is more important to him than that his other parent also gets the one that his she wants.
This has to be non-negotiable, otherwise he'll take it as a licence to overrule you in any number of things in the future.