I cross-posted this to reddit's /r/polyamory. My date was a redditor. He had his wife create a throwaway to respond to me. Long story short, they came at poly from the swinging/nonmonogamy side. This rule is a remnant of navigating poly from that end of things, apparently.
She says she's open to renegotiating it/cancelling it if everyone agrees, and that she's feeling compersion that I went out with him and liked him so much.
So, that's positive!
In return, I wrote her explaining why vetoes suck for me personally, and what I'm wary about, and the pitfalls these kinds of rules tend to fall into. I also mentioned that it was obviously her relationship and her rules, and that I respected it even if it wouldn't work for me. And that if she wasn't willing to compromise, I could bounce and wouldn't be offended.
So...hopefully it works out?