I would think that the guest could say, "Oh, I'd love to, but I think I'd better say no--my cousin is visiting from out of town, and I can't really leave them home alone."
Any generous host would assume that an invitation to the cousin would be welcome.
I've never agreed with the "it's not rude to ask" idea. There are situation in which it truly "doesn't hurt to ask," but they're never about someone else's hospitality. (They're stuff like, can I get an extra pat of butter at the restaurant; do they have this in a size 8 in the back; etc.)
Goodness - didn't we just have one ( or more ) long threads about whether or not "I can't go" need more explanation. One of the frequent assertions is that if you don't want to go somewhere, you should say something vague like "I'm sorry I can't make it due to <XYZ>" and a polite asker will not probe for more details but accept that answer as final.
I would think that the potential guest would choose wording that would indicate the message -they- want to send. The guest has all the power in terms of choosing what they want to communicate. They can give detail--or not--as they choose.
And enthusiatic or wistful, "I wish I could, but my cousin is visiting, I can't leave them by themselves" would say to me, "You could invite my cousin!" (or sister, in this case)
A "My cousin is visiting, and we have plans for that night" would make me think that an invitation might be declined, and I might not even offer.
And "I'm sorry, but I'm busy" would make me say, "Oh, you're busy, rats, another time then."
And of course the extra explanation is not *needed*, but it's a very useful tool. One that the guest can wield--or not--as she/he wishes.
Potential Guest: "I can't come because my cousin is in town"
Potential Host: "That's OK! Cousin can come too!"
50% of Guests think "I'm so glad <host> asked, we really wanted to come!"
50% of Guests think "Darn <host> is rude, I really don't want to go and now I have to find _another_ reason!"
But the guest who really didn't want to come could have chosen a different answer that would stop the invitation. Or, now the guest can simply elaborate on the _same_ reason, "Oh, no, I'm sorry, we have plans for that night already--sorry I wasn't clear."
Again--the guest has all the power. (Maybe they don't have the skill--in which case, they will have to simply come up with another phrase to use when the invitation is extended to the cousin--and hopefully they'll build on that skill.) And it cannot possibly be rude to extend an invitation!!