I think you have a really interesting situation BUT the goal of simply convincing her leaders is a bit weak. You've got this really cool post-apocalyptic setting, but all the action could take place in one room. There is the potential for SO much more!
Easiest option - she runs away and they start hunting her. Now that's more like it! Obviously whilst out on the run, she discovers a community on the edge of society that do things their own way, everyone can live the life they choose and they take her in. Chances are she meets a guy there and her views start changing, but then her presence there threatens this small community... etc etc.
From a logline perspective though it's pretty good.
In a post-apocalyptic future where reproduction is only for "the worthy", after announcing she doesn't want children, an independent young woman, chosen for reproduction, becomes a fugitive with a bounty on her head as she seeks to live her life as she chooses.?
Given how much is currently going on in the world about what women choose to do with their bodies is their choice, the #MeToo movements, etc etc. it's the perfect time for a film like this.
Good work!