Because he wants to make his dad laugh. Simple as that.
Maybe Adrien says, 'Dad, why is Hawk Moth always so gloomy? Because he's in a bad'mood' (as in his evil plans).'
Maybe the joke is 'Why is Hawkmoth always so stressed? Because he has too many akumas to manage!'.
There are multiple consequences. Firstly, his relationship with his father will be forever changed. He'll no longer be able to trust him in the same way. Secondly, his relationship with Ladybug might be affected. She may be more cautious around him, not knowing if he'll side with his father. Thirdly, Adrien will have to face the public's reaction if the truth ever gets out. He could become a target or be shunned. All of these aspects make for a very intense and emotional storyline in fanfiction.
He might tell his dad the cat pun fanfic because he found it really funny and wants to share the humor. Maybe he read it and just couldn't stop laughing at all the clever cat - related wordplay.
Maybe his dad gives a small smile and a nod. He could be a bit reserved in his reactions but still show that he appreciates Adrien's attempt at humor. He might say something like 'That was quite clever, Adrien.'
He might be shocked at first. In many fanfictions, Adrien would feel a mix of disbelief, anger, and sadness. He has always known his father as a strict but loving parent, and finding out he's Hawk Moth, a supervillain, would completely shatter his view of him.
A good fanfic about their bonding should have well - developed characters. The author needs to show the different sides of both Adrien and his dad. For example, Adrien's dad could be strict but also have a soft side when it comes to his son. It should also have a believable plot. Maybe something as simple as a day at the park where they have a meaningful conversation. The dialogue is key. It should feel natural, like a real father - son conversation.
It could be Gabriel Agreste. He has been a suspect in the original series for various reasons. He is often absent and has a lot of secrets. So in the fanfiction, it would make sense if he was finally revealed as Hawk Moth.
One possible plotline could be that they initially struggle to adapt to each other's lives. Adrien, in his dad's body, might have to deal with the pressures of running a business, while his dad in Adrien's body has to handle school and social life. This could lead to a lot of comical misunderstandings.
Well, perhaps the fanfic is about a world where cats can talk and Adrien is sharing with his dad all the funny cat - related puns from this fictional world. For example, the cats might make puns about their favorite fish or the warm spots they like to nap in.