I vote with Jan.
I hate family situations like this. To me it begs the question, why do so many people with dogs, big or otherwise, think everybody should love having them around? Add to that, why are someone else's little darlings and their bad habits supposed to have free run in someone else's space? Your cousin could get someone to take care of the bothersome critter while they're gone. I can't have animals in my apt. so that's an easy one for me. My kids weren't allowed to have things like food and drinks all over the house and so I wouldn't allow a guest's child to carry milk or other drinks around either. Well, maybe water. My guess is that I'd tell my cousin how it is, when they arrived I'd enforce the rules I'd set, and my cousin would soon go home and probably wouldn't want to come back very soon. I often had the opposite problem: At home my kids were expected to keep food and drinks in the kitchen or other appropriate spot and when we were away from home others would encourage them in what I considered bad habits. Good luck keeping your home your home while not alienating family. I hope it's possible for you all to have a good visit and nobody get offended, upset, or mad.
Cheating is an option. . . . Cheaters never win and winners never cheat, but smart knitters who want to retain an iota of sanity do, cheerfully. ~~Kory Stamper