Girlfriend wants me to say I’m Jewish, but doesn’t want me to convert?

Hi,
I’ve been in a serious, committed relationship with a religious Jewish girl for quite a while now. Although my dad’s parents were born Jewish and my last name is an identifiable “Jewish” name, I am not Jewish by birth, nor was I brought up Jewish in any way. As things have gotten more and more serious in our relationship, we started talking about the possibility of having kids. When this happened, my girlfriend said that she really wanted to raise our kids Jewish, and I agreed. But I also said that I’d like to convert to Judaism. I’d been looking into it more and more over the course of our relationship and getting really into it, and genuinely think that I’d like to convert to Judaism. Plus, if our kids were going to be raised Jewish, I’d like to be able to connect with them on that level and not try to raise them in a context of which I am entirely unfamiliar.

So this is when it gets a little bit confusing. She doesn’t think that I should convert, and often publicly just insists that I’m Jewish. She wants me to pretend! When I go to her family events like celebrating shabbat, or going to her nephew’s bar-mitzvah, or things of that nature, she tells everyone that I’m Jewish in advance, and then they just assume. This is uncomfortable in two ways for me:

1. Eventually, at these religious events she brings me to at synagogue or through hillel on campus, the topic obviously turns towards religion. It feels weird and uncomfortable to be introduced in a certain way as a “Jewish Guy”, and then very clearly not know anything that would be expected of such a person. I feel like I’m lying to people and I don’t want to.
2. Whenever I actually try to be open about not knowing much about it, and ask questions to the people around me and try to learn more about it, it gets somewhat uncomfortable, because she’s already gone and introduced me and told everyone I’m Jewish.

How do I handle my situation? My feeling is that at some point I either need to convert or not, but pretending that I’m Jewish when going to events is not something I should be doing. But I wanted to get the opinion of people on this subreddit because I honestly don’t really know what is or isn’t ok / expected behavior. What should I do?