I can understand the annoyance about the bf at the party. If he were that serious of a bf, they would have met him already. Bringing him to the party with the purpose of the family meeting the fiance' sets the bf up as equivalent. That makes it different than a 90th birthday party. Also, the cousin's bf is local, and the fiance' isn't. She can introduce her bf to her immediate family some other time.
I don't think that it would set him up as equivalent. This was the OP's engagement party; her fiance was guest of honor. I presume he got gifts and toasts. The boyfriend would just have been an ordinary guest. That's not at all equivalent. It would not upstage anyone or anything.
And there was nothing about wanting to introduce him to her immediate
family (she may already have done that anyway). I agree, some other time would be fine for that. The point was for him to be able to meet the extended
family. Especially the OP, in fact, because she lives overseas. That's why it made sense to ask if he could come to this party; when else to do that other than at an extended family event? In fact, it's so common that it comes up from time to time in the form of someone criticizing the guests of honor who complain at feeling like they have to share even a tiny bit of attention. And this wasn't even the wedding.
The question someone asked above about whether he would have been included if he had already met the family made me think of something else: how many people would it take before the OP's mom wouldn't have thought it upstaging? I mean, suppose that his boyfriend had already met some of the family, at least Cousin's immediate family and some of the others. Does he have to have met more than half of the other guests or something before it wouldn't be "upstaging" to have him there so he could meet the others? 30%? 80%? I think that it was fine for Cousin to ask, and I bet it wouldn't have bothered the OP if she didn't dislike her (which we can all relate to! But still.).
Pooky582 makes a good point, though. This and the dress issue are red herrings.
The question is whether there is a polite way that the OP can fire her cousin as bridesmaid. And I think the answer is no. To the extent the dress and party issues are relevant, it's only because they are the only things Cousin has done since the OP asked her to be an attendant, and the question is whether they are bad enough to justify firing her. There seems to be pretty universal agreement that they aren't. We can understand the OP's annoyance, but it's not like she slept with the groom or killed the OP's puppy or something. Way disproportionate.
So it comes down to having to own her real reason for wanting to get Cousin out of the wedding: she doesn't like her.
That would have been enough not to ask her in the first place. But she did ask her. Rescinding that invitation would be a huge deal -- and as she is a relative, one with awkwardness for other family members, too -- and to be justified must be based on something other than regretting having asked her.
OP, consider that Cousin being a bridesmaid is probably not going to affect you much on your wedding day; she won't be a point of focus -- but there is no way you can kick her out without that
becoming a point of focus, both for you and for the family. You really don't want to be having that as one of the major memories of your wedding (for you and the family guests). If you fire her, it will spoil your own pleasure because you will be second-guessing yourself every day before the wedding and perhaps beyond.
Right now you are the gracious and forgiving one who invited Cousin to be an attendant even though she had behaved badly to you in the past; you can stay there, or you can become the one who humiliated Cousin by kicking her out of the wedding for no good reason (and, some might even wonder, planned it all along and only asked her in the first place just to be able to fire her).
Don't take that risk. You're never sorry when you behave as your best self, not do the least you can get away with. You know what the right thing to do is. I promise you won't regret it. And you'll be able to stop thinking about her (and how annoying she is!) all the time and focus on your wonderful wedding!