It's a bit hard to say for sure. Both Vegeta and Goku have their moments of strength. Goku has some unique powers and techniques that give him an edge at times, but Vegeta is no slouch either and constantly works to catch up and surpass Goku.
In this fanfiction, Goku has the potential to be stronger. He has mastered different forms like Super Saiyan and beyond. However, King Vegeta is not to be underestimated. He has his own Saiyan pride and power. If it comes down to a fight, Goku's adaptability and continuous growth in power might give him an edge, but King Vegeta's experience and strategic mind could also make it a very close battle.
His power would develop more in line with Saiyan traditions. He'd probably have a stronger base power due to the intense training environment on Planet Vegeta. There would be more emphasis on raw power and combat skills from a young age.
Well, he would probably be a very different person. He might be more in tune with the Saiyan warrior culture. He could be more aggressive and less empathetic compared to the Goku we know. He would grow up learning the ways of the Saiyans on Planet Vegeta, which means he'd be trained to fight and conquer from a very young age.
Bulma could be involved. Since she has a relationship with Vegeta and is friends with Goku, she might try to mediate between them when Goku snaps at Vegeta. She could be the voice of reason, trying to calm both of them down and make them see each other's point of view.
In general, it's a bit of a toss-up. The strength of Manga Goku and Anime Goku can vary depending on the specific arcs and battles. Sometimes, it might seem like one is stronger in a particular situation, but that could change in another.
Yes, in the vast world of fanfiction, there are all kinds of creative and out - of - the - box stories. Some fans like to explore unusual relationships and scenarios, so it's possible that there is such a fanfiction out there. However, it goes against the original canon of the Dragon Ball series in many ways.