Oh, I know how hard this must be for you. Unfortunately, I didn't learn about carbs in time to change my DD's way of eating. I agree with your sense of urgency here! My DD is 21, living away at college and morbidly obese - it's too late for me to change the way she eats... I can just show her my success as an example.
At 16, I think you still have time to make some changes to get her on the right track. Do you have any long weekends or summer visitations coming up? If so, then you could try weaning her from carbs over a few days (assuming there isn't a medical reason here). Encourage her by example to go lower carb, not very low carb. So, only have diet sodas or pitchers of tea made with sugar substitute, make bacon and eggs for breakfast instead of cereal, that sort of thing. When I started lc'ing, I just started lowering my carbs... cut them in half initially, then every week just dropped them lower. It worked for me and I think it might be good way to start for a teenager.
Secondly, does she like to shop? Most girls that age do, so take her shopping and get her to open up with you about her clothing size. Try to gently guide the discussion so that you can get her to discuss how she feels about her size. Don't be pushy, though, just let it come naturally and let her reach out to you.
I don't have to deal with an ex, so I'm not sure quite what to do there. Somehow, though, her mom needs to learn about low carb. Even if the mom doesn't need to lose weight, she can help her daughter by learning about it and fixing better meals at home.
Feel free to drop by my journal if you want to bounce around some ideas. I've tried a lot of things with my DD, some worked and some backfired in my face.
I hope you find a way that works for your SD. Good luck!