Letters to the Editor
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Step One: Grow a Backbone
Step Two: talk to your husband about his over-enthusiast invitation, and how its results cause stress in your life.
Step Three: decide on the united front that the two of you will present to the world (not only pushy cousin.)
Step Four: implement.
No matter the grandiosity of the invitation, NOBODY jumps in a pool when the owners of the home are entertaining guests, unless s/he is extraordinarily stupid, or extraordinarily uncaring about anything but his or her own pleasure.
People are raised differently, that's not news. My husband used to be very blaze about saying that we'd arrive at someone's house at such and such a time, and then we'd be leaving at that time--and it was over an hour's drive away. He'd say, "Nobody ever minds, they never say anything!" The idea that people were being polite by NOT pointing out his chronic lateness had literally not occurred to him. He, the son of the woman who nearly missed getting on an international flight because she needed some water to drink on the way to the airport, and couldn't find just the right container for it.
Your husband's cousin could be the same. "They invited me to come at any time, and haven't said anything!" Granted the jumping in when there were invited guests there was way over the line, but why on EARTH did you NOT tell her, "Oh, gosh, Cousin Suzie Q, we have guests in the backyard today, please come another time."?
Back to step one.