What you are describing is very common - most beginners find the hardest part of the pull-up to be the first few inches of the ROM. Though I'm personally not a huge fan of lat pull-downs, it sounds like you are doing most of the right stuff. It's just a slow process for women. And as you mentioned, dropping more body fat will definitely help. I wrote a guest post for MDA about pull-ups a few months back that you may have seen. Here's the link in case you missed it: The Importance of Pull-up Bar Training | Mark's Daily Apple
And here's another article I wrote that's specifically directed towards females: Pull-ups For Women
Be patient and keep training hard!