Why did you dislike Carnival?
If it was due to large crowds and poor food, the RCI would also be a bad selection.
If you want good food and amenities but want your kid cared for, how about Disney?
I haven't sailed Celebrity for 20 years and have never sailed HAL, but from what I have read, I think Millie would be a wonderful cruise.
If you want to island hop, enjoy good food and service, and leave your daughter to fend for herself, I would recommend Seven Seas Navigatror sailings out of San Juan. Ft Lauderdale and Miami sailings will always have sea days while San Juan sailings can have a new Island each day.