It sounds like you miss the positivity you once saw in her. One option is to remove attention except when you see what you like, and then turn the spotlight on and shine. Essentially retrain her to be the person you are willing to be with. Limited eye-contact, distracted "mmm-hmmm's" when she's being a bore, and then when you see glimpses of what you liked in her, full eye contact, and a big smile and nod. She'll start giving you more of what gets her what she wants. Especially if life requires you to be around her now, such as a co-worker, or wanting to be around her sister without awkwardness.
You get to chose your limits. The reality is that she's not going to know your weight unless you tell her, or what you're eating, or whether you've lost, or how you're working out. Just set boundaries. If she insists on bringing it up, say "I've decided not to discuss my weight, food, or exercise with anyone but my doctor for the present." If she presses, repeat the same sentence until the message has been received.
Who cares what she thinks about whether you can lose weight, since you're doing it? Talk about the weather, or how she's decorating for the holidays, or a movie you've both seen. Or do things with her that don't involve conversation--go to a concert together.
People can be peculiar. You could probably salvage this relationship if you decide to, but keep it light, get in, get out when you've had enough.
I knew a woman once whom I had to deal with; worked with her husband, lived in the same building, our kids were friends. She looked at me once in shock when I'd been losing weight and said, "How much do you weigh?" I told her, and she said, "I have to lose weight." Woops! Message was clear--my job was to be fatter than her, and I'd screwed up. She defined herself, but I just laughed inwardly and moved on. Some people are true friends, and some are just acquaintances, and won't ever be worth the time to get to the bottom of problems like this. She sounds like this kind of person, but also probably not worth the energy she would take if the relationship turned antagonistic.
Just create a little distance each time she offends. "Nope, can't see you. Busy. Gotta go. I have plans that day, sorry. Something in the oven, bye." Etc.