Hey there,
I think that your friends probably see more in you than just the fat bother. But maybe you don't? That would be understandable regarding your story and how your family treated you but we are born into a family and don't get to choose them, so we all have to live with the one we got.
Obviously you are not happy with who you are and how your life is going right now. I think you took a great first step with registering here and trying to lose some weight. Not for them but for yourself and your self esteem.
If you feel like the only one with problems weight related or else, it could be helpful to look for friends who also have to struggle. The Internet is such a great opportunity to find others who also have issues to confront, maybe even the same.
As for your daughter - you are her mom, of course she will love you! Kids don't judge their parents by appearance, at least not when they are so young. Your daughter doesn't need a skinny mom now but one that is happy with herself and her life. So that is where you should start
How is your relationship with your ex? Can you talk to him about it? I guess for your daughter it would be best if you both could decide on a common style of raising her, not because she won't love you but so she doesn't get confused and has some rules that are reliable for her.
Last but not least I do think as well that you should get professional help, if you want to. Not all therapists and therapy styles are the same. I personally went to therapy as well when I was younger and it didn't really fit me so I thought it's nothing that could help me. Now I am seeing a therapist again and this time it pretty much changed my life and helps me a lot.
Maybe you could also get some family counciling or something.