Is being with a fat girl super important to you? If so, don't waste your time and your girls time by being "quietly optimistic". At the very least, you should be openly optimistic about it. If she wants to run marathons one day or something, you're probably not the right guy for her.
Is being with a fat girl kind of a plus but not necessary for a fulfilling sex life? Then stay with her. She seems like a pretty chill girl. But you must be prepared to face the fact that she will never get close to nearing 200lbs.
From my experience with dating a girl like this, you're gonna be lucky if she puts on 20 pounds in the next decade or so.