You've already pinpointed the issue:
> She's single and a few years older than you (late 30s). She's from a cultural background where she would be regarded as being “on the shelf”. Looks-wise she's unlikely to turn heads instantly, but she is not ugly.
Women in current_year have been brainwashed into viewing men as the enemy. If she's late 30s and childless, she likely resents herself for missing her window and is projecting her self-hatred on to you, the enemy. You are the one who is forcing her to live in an oppressive Patriarchy. You are the one who has robbed her of her self-agency. You are the one who has stolen her dreams by forcing her biological clock to cause her to be so neurotic. Of course, we all know that's bollocks, but people infected with the mind virus are not rational.
When I was a fresh graduate working on a mine site many years ago, there was a woman in a much more senior and more important role than me (superintendent level) who was the most bitter, sour-faced cunt I have ever met, even to this day. I didn't know the bird and hadn't spoken a single word to her, but she would give me the stink eye every time we passed in the corridor. Me being an insecure young 20 year old eventually got fed up with it and asked her point blank one day what her problem with me was. She sneered at me, refused to answer, and went and complained to my boss who called me in for a chat. He told me that I'm not the first male that she has had a problem with and that it's a known issue but he couldn't do anything about it because the company was actively looking to hire women and that they wouldn't circumvent their KPIs for something like this.
In hindsight, I was incredibly lucky that my boss was so cool and rational about it, because it really opened my eyes to the nature of the times we are living in. It was kind of my own personal Great Wokening because it wasn't long after that I heard Obama publicly pushing the wage gap lie and it just didn't mesh with my own perceptions. My lived experience, per se. I knew there was absolutely no way that women were being paid less for the same job when management had KPIs to hire women. What kind of manager is going to be underpaying women when they're incentivised to hire as many of them as possible? It's all part of the soft cultural coup that has been waged by feminists for the last several decades, and this woman's attitude is very likely a result of her being convinced that she's oppressed and you're to blame.
Long story short, the best thing you can do is what you've already done: return the could shoulder. Don't speak to her unless you absolutely have to for work purposes, and even then just keep it cold and professional. Never allow yourself to be in a room alone with her and if you do, turn on the recording on your phone. Above all, do not, under any circumstance, confront her about it. I'm not sure what industry you're in, but given the modern culture, she very likely has the upper hand in any HR complaint.