Welcome to the forums. I think the feelings you're encountering are perfectly natural. And I think the right thing to do is continue to talk about these feelings with your SO as openly and honestly as you know how. I think you're right to want to meet his new paramour, and I'm a little perplexed as to why she is saying "no". Possible red flag there.
Perhaps she thinks she will threatened by you. It's hard to say. What does your SO say about her refusal to meet you?