Chances of daughter being refused entry to U.S

I know this maybe a n unusual question with no definitive answer but I wondered if anyone had any idea of the chances of my daughter being refused entry to the U.S? Basically I had a short relationship with her father whilst travelling in the U.S and married him pretty quickly in California. I fell pregnant and left him to return to the U.K whilst I was pregnant. Roll on 12 years and we have since divorced and I never saw him again after leaving the U.S as he turned out to be not a nice person. He has never met his daughter and we've only ever had the odd letter from him.
I am now married to another U.K citizen and I have 2 children with him. We are wanting to take a holiday to the U.S but have been advised that as my daughter has a claim to U.S citizenship she will have to travel to the U.S on an American passport. The problem is that we don't know the whereabouts of her father and so are unable to obtain the required paperwork for this to happen. What are the chances of immigration being aware of her potential claim to U.S citizenship if she travels on her British passport?
]If you've managed to read this far then thanks
]I hope it makes sense.

A US citizen cannot be refused entry into the US, even without a valid passport. What does happen from time to time is that a US citizen without a US passport is refused boarding by the airline, but in your daughter's case this is unlikely to happen.

In any case, I would strongly recommend that you get proof of your daughter's US citizenship sooner rather than later. One day she (or someone else in your family) may wish to move to the US, or work there, and the more time goes by, the more difficult it will be to prove her claim to citizenship.

It is my understanding that a US citizen has to enter the USA on a US passport. However, in practice, the immigration officer will rely on the place of birth to make that call.
If your daughter was not born in the US which is what I infer from your post, then her place of birth will be somewhere in the UK. Assuming she will be travelling on a UK passport, it will be impossible for the immigration officer to establish any possible claim to US citizenship based on the particulars on the UK passport. Hence, there is no issue.
Furthermore, a claim to citizenship is not the same as citizenship itself. Your daughter was born in the UK and is a UK citizen. She is most likely not automatically a US citizen, purely based on the fact that her father is one.