It may be "my house", "my dog", "my party", etc but I wouldn't feel entitled to use any of them to exclude or hurt anybody's feelings. I'd love to see kids in family parties but not at serious ceremonies.
Your cousin's been very mean & petty IMO especially if you've always got on. Even if you haven't, this is not the way to go about it, using her dog.
She/he's been really reckless by being prepared to let loose a dog among his/her family reunion that has a history of biting. Has he/she considered the consequences of it for her/him self, others or her dog! They could be sued if the dog bit someone at the party or relationships could suffer. To be quite honest I wouldn't go because I wouldn't want to be around such a dog and I wouldn't be happy if my family took such a risk. It simply isn't worth it, is it?! I'd thank her for her kind invitation and apologise for not being able to go. If they probed, I'd say that I have a serous problem around territorial dogs.