How can I encourage my daughter to be healthy and not get fat?

For context, my daughter is 6, and I share custody with my ex-husband.

Over the past few months, I’ve noticed that she's been getting fatter. While I don’t allow her to eat junk food, it seems that her father lets her eat whatever she wants. He denies it, but I’m starting to suspect that he might be doing this intentionally to make her resent me. After all, I'm the one keeping her on a diet and telling her to go to the gym while he allows her to laze around.

And sure enough, my daughter is now insistent on spending more time with his father. What can I do to make my daughter healthy without making her hate me?
How can I encourage my daughter to be healthy and not get fat?
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