Really the only thing that needs to happen is to exploit what both Characters are about. Godzilla is an anti-hero who fights to keep nature in balance and will come to blows with anyone threatening that balance. Kong is a sympathetic creature who fights to protect the weak.
We see in Legendary Goji that he's been around for awhile. He's seen the dangers that the world has to offer, This includes the Humans.
Kong, Is still relatively young, his encounter with humans has mostly been a positive one despite a few bad apples, He's capable of figuring out which humans specifically are bad and which ones are harmless and he tends to give them all a chance to show their true colors before acting out at them.
We know Monarch has been on the island working on some new tech. The very threat of a new weapon should be enough to put them on Godzilla's radar. Especially if testing causes restlessness of the creatures on the island.
Kong, Having been around them for awhile now, doesn't see the harm in their presence as they tip toe around him. (Monarch would know better than to openly blast the island by now.) It may even be that a specific human or group of humans has befriended Kong.
This is really the only set up you need.
Godzilla shows up, sees what monarch is doing and begins to take them out. He's destructive and Kong doesn't like it.
Godzilla gets the upper hand with his experience on their first encounter but Kong survives. The Humans then employ their tech to help Kong in his next fight and Kong fairs much better the 2nd time around. Godzilla refuses to give in and pulls out all the stops. In the ensuing chaos The humans unleash their new weapon and it gets a few of Kongs friends "Be it Creature or Human or otherwise." killed and Kong realizes the dangers of the weapons the Humans possess. Just as Godzilla is about to atomic breath the Monarch base, humans and all. Kong steps in and destroys ONLY the tech in front of Godzilla. Leaving the humans alive but defenseless.
Seeing the humans are no longer a threat and with a look in the eyes of Kong saying, "I'll keep the humans in check you don't have to worry about them.", Calms Godzilla's rage and he leaves the island with a new found respect for Kong and a promise to be back should their be any more monkey business.
So you leave with the knowledge that while Godzilla is King of the Monsters, Kong is King of the Humans.
Ps:
Regarding their ancient rivalry, I'd play it out something like this.
Kong and Godzilla species are both Alpha Predators. But they usually take turns between reigns. So it's not often that one of the members species is roaming the surface at the same time as the others. But it has been known to happen in the past that when the two happen to be awake at the same time, they don't always get along. Despite both being protectors of life, They have opposing views what exactly that means.
Angering the Godzilla fan base one take at a time
