The start of the 'hercules killing family story' is rooted in the animosity between Hera and Hercules. Hera, being the wife of Zeus, was enraged by Zeus' affair with a mortal which resulted in Hercules. So, she used her divine powers to make Hercules go insane. This insanity was the trigger for the tragic event of him killing his family.
Well, in the 'hercules killing family story', Hercules was driven insane by Hera's jealousy. He didn't realize what he was doing when he killed his family. It's a really dark and sad part of his story. After this horrible act, he was filled with remorse. And that's why he had to do those extremely difficult labors to atone for his sins.
The Disney story of Hercules has some basis in the real myth. The core idea of Hercules being a heroic figure with extraordinary strength is there. But Disney simplified and modified a great deal. In the real myth, Hercules had to complete the Twelve Labors which were often quite dark and difficult. Disney took that concept and made it into a more upbeat and less gory adventure for kids. Also, the way Hercules' character is developed in Disney is different from the more complex and often flawed character in the original myth.
Their interaction would be tense at first. Young Kratos, full of energy and unbridled aggression, might see old Kratos as a threat or an obstacle. Old Kratos, on the other hand, would be more reserved and cautious. But as they start to communicate, they would find common ground in their shared past and their connection to their family.
One of his main adventures was the Twelve Labors. He had to perform tasks like slaying the Nemean Lion, which had an impenetrable hide. Another was cleaning the Augean stables in a single day. He also captured the Erymanthian Boar.
Kratos might win. He has fought and defeated many powerful gods in his own world. He is not only physically strong but also very strategic. He could analyze the Hulk's movements and find his weaknesses. The Hulk, while strong, is more of a brute force fighter. Kratos could use his Blades of Chaos to keep the Hulk at a distance and then strike when the opportunity arises. Also, Kratos has faced opponents much larger than him before and has always found a way to win.
One possibility is that it could focus on a mentoring aspect. Old Kratos, with all his wisdom and experience, could guide young Kratos. Maybe he'd show him different combat techniques or how to better control his rage. It could also be a story of self - discovery for young Kratos as he sees his future self and realizes the potential path he could take.