Traveling as a Dual National

My child is a dual US UK citizen and we will be traveling between the countries. I put in her US passport number as we are going to the US first and called the airline and asked if I can enter her UK passport as well, and they told me she will need a visa to enter the UK and got quite ass-y with me and then said they don’t give advice except she should travel on one passport (?!).

Any advice??