Thank you everyone for your support and input.

I should add that in our field of work, it's very normal when you're looking at a company you like to find someone on LinkedIn who works there and to ask for a reference, even if you're applying for a full job and not an internship. Very often, they'll want to skype with you for 30 minutes to an hour, review your publications, and from there it's not a big deal. That's part of why I'm so annoyed that my cousin is acting like this. These internal references aren't full-blown recommendation letters by any means.

My SO doesn't actually need her reference. He can find someone on LinkedIn to do it, or just not have one--internal references aren't required. So not having her reference doesn't particularly affect him. I'm more upset on principle, that literally she's less willing to help me/us than most random strangers or distant contacts on LinkedIn.

It's true that she knows I have problems with my job, but she didn't always. Even when I was really excited about the work my group was doing and hadn't shared much about all the dysfunction, I'd get her random texts that I should come to her company. I never confronted her about it, which I kind of regret. I always told myself that she was just "being nice" and "trying to be helpful" etc. Then I told myself not to burn bridges in case I wanted to apply to her company. But she really has always had a high opinion of herself and a big head.

I'm debating whether or not to confront her. Honestly, there are a few other details here I haven't been comfortable including in this post, that unfortunately really affect the context and are part of why I'm so extremely angry. I'm really tempted to write her off as "less willing to help me than most total strangers" and stop talking to her.