She should be more active but it might just be shyness at this stage and have nothing to do with her energy - my kids would never play at the playground at that age. How is she at home?
I missed that about the juice being only 2oz - seems totally reasonable to me. I think you are doing a great job and maybe thinking about what you can add to her diet might do more long term (rather than take away). Maybe you are right that LC kids are the pudgy ones when young and then they thin out?
At two, it's unlikely that you have any control at all over the situation and she will just thin out or not as she gets older. If it's any consolation, I was heavy since 4 and it didn't prevent me from dancing my whole childhood and in my 20's. Now I run long distance races (including 2 half marathons) and am very healthy. Fat is never the whole picture, KWIM?