Well, Pierre Menard is definitely fictional. Usually, when we talk about him, it's within the context of a fictional story or narrative where the author invented this character for various purposes.
Alexis might confront Beckett directly. She could say something like 'I think you're cheating on me. What's going on?' This is a very straightforward approach.
I don't know who Alexis is in this story without reading it. But it's likely that Alexis is either the main character or an important figure in the events that unfold in the 'creepy subway' setting.
A lawyer could be an important character. This lawyer takes on Alexis's case, believing in their innocence. The lawyer has to battle a corrupt legal system or powerful opponents who want to see Alexis locked up for good. The fanfic could show the lawyer's strategies and how they build a case in Alexis's favor.
Perhaps there have been some rumors or whispers in their fictional world that have reached Alexis' ears. Even if they are baseless, they can plant the seed of doubt in Alexis' mind, making her think that Kate is cheating.
She could be defensive of Beckett. Since Alexis probably has a good relationship with Beckett, she may try to convince Castle that he's wrong and that Beckett didn't mean any harm. For example, she might tell Castle about all the positive things Beckett has done for her.
I'm not sure who Alexis Rhodes specifically is in this context as the name is rather common. There could be many fictional or real people named Alexis Rhodes. Maybe it's a character from a particular book or a personal account that I'm not familiar with.
Well, without more context, it's difficult to define Alexis' secret story. It might be related to a family secret that Alexis has been keeping. Maybe it has to do with an inheritance or a long - lost relative. Or perhaps it's a story about Alexis' first love that was full of unexpected twists and turns and they've chosen to keep it a secret.
Kate might get defensive right away. She could say things like 'How could you think that of me?' and be angry at Alexis for not trusting her. This is a common reaction when someone is wrongly accused in a fanfic or in real life.