I'm not sure if there is an actual fanfic with that exact plot. Fanfic topics can vary widely, but this seems like a rather unfaithful and out - of - character concept considering Chi Chi's strong bond with Goku in the original series.
Maybe in this fanfiction, Goku realizes that he and Chi - Chi have very different personalities. Chi - Chi is often more concerned with family and a stable life, while Goku is a free - spirited adventurer. This clash of values could lead him to decide not to marry her. For example, Chi - Chi may want Goku to stay at home and help raise Gohan, but Goku is eager to explore new planets and fight powerful enemies.
One reason could be his pursuit of stronger opponents. Goku is always eager to test his strength and find new challenges. In fanfiction, this might lead him to leave Chi - Chi to travel the universe in search of the toughest foes. He might feel that staying at home with his family restricts his growth as a fighter.
One idea could be that Goku forgets their anniversary. Chi Chi is initially angry but then decides to use this one - week period to remind Goku of their relationship. She sets up various challenges and tasks for Goku related to their past memories together. As Goku goes through these, he rediscovers his love for Chi Chi in a deeper way.
Power struggles can also be a common theme. Dark Goku is often depicted as very powerful. Chi - Chi, on the other hand, has her own strength in terms of her determination and will. The power dynamic between them can create interesting scenarios in the fanfic. There could be moments where Chi - Chi has to find ways to stand up to Dark Goku, or Dark Goku might try to use his power to manipulate or control her in some way.
Love and family are very common themes. Since they are a married couple and have a son, many fanfics explore how they navigate their relationship while also raising Gohan. For example, in some fanfics, Goku tries to be a better husband and father by spending more time at home.
Goku would likely be extremely excited. He has always had a longing to know more about his Saiyan heritage and his father. Meeting Bardock would be like a dream come true for him. He'd probably ask a million questions about his past and Saiyan culture. Chi - Chi, on the other hand, would be a bit more reserved at first. She'd be polite but also a bit cautious, as she doesn't know what this Saiyan from Goku's past might be like. But as she sees Goku's joy, she'd warm up to Bardock.
Family is a common theme. Since they have a family in the Dragon Ball series, fanfictions often explore how they raise their kids. Another theme is love, of course, showing the depth of their relationship.