I can't use my BRP anymore because it is invalid. Can I travel just using my US passport without an issue?
As a citizen of the UK, you have an unrestricted right to enter the UK. The problem will be proving that right if you need to invoke it explicitly. In other words, you might have a bit of a delay entering the UK on your return.
When you enter the UK, you could either show your US passport and say nothing about your UK citizenship, or you can mention that you are a citizen of the UK and show whatever evidence you have of that fact, such as the certificate you received at your citizenship ceremony.
If you do the first, they might somehow see that you are a citizen of the UK and ask you about that. Or, if they don't notice that you're a citizen of the UK, and they also decide that you look like someone who might try to violate immigration law, they might decide to refuse entry. At that point you would have to claim citizenship of the UK in order to avoid being removed.
It's probably better, therefore, to start by showing your US passport and UK citizenship certificate and explain (if needed) that you didn't have time to get a UK passport before your trip.
My father believes that they will only let me stay for 90 days at most.
This is incorrect, even if you use your US passport to enter without identifying yourself as a citizen of the UK, and even if you receive a standard six-month "leave to enter" stamp noting that employment and recourse to public funds are prohibited. Once you've attended the ceremony and received the certificate, you are a citizen of the UK, and your presence in the UK cannot be restricted just because you've entered with a different country's passport. Any stamp purporting to impose such a restriction would have no effect.