Monsterverse Idea

I'd have Godzilla vs. Kong take place before King of the Monsters. Here's how I'd have the universe play out:

Godzilla vs. Kong

The public fears the threat that Godzilla poses, and so the US military looks to Skull Island's Kong so the two fight to the death. They plan to lure Godzilla there with a nuclear bomb, them detonate it while he and Kong fight. Among the soldiers sent to Skull Island is Ford Brody.

The military goes with this plan, and Godzilla arrives at Skull Island. Kong, having grown to roughly the same height, sees this and the two kings duke it out. Neither is able to overpower the other, with both gaining great injuries. The fight decimates the landscape and causes some military helicopters to crash and land on the island. Ford Brody and some other men are caught in the fight. They are rescued as the two monsters are severely injured. Brody escapes as the bomb is detonated, ravaging the island. Believing the monsters to be dead, the military leaves, but unbeknownst to them, both have survived.

King of the Monsters

Satellite technology shows that a meteor is on a collision course with Earth. Monarch examines cave paintings showing that a similar meteor hit Earth in the prehistoric era. They discover that the meteor is a vessel for Ghidorah, who travels to Earth every once in a long time. In the incident depicted in the cave painting, Rodan and Mothra awoke to stop the giant space dragon. Some question why Ghidorah has not eradicated humanity, to which Monarch says that Godzilla and other earthly monsters stopped him, showing the cave paintings from the post-credits scene of Skull Island.

The military attacks the meteor, which tanks their weapons and hits the planet. Ghidorah awakens and decimates the military, continuing to ravage the planet, with Skull Island being his first target. Godzilla, healed from his fight with Kong and still on the island, awakens to fight Ghidorah. The two meet and clash, but the King of Monsters is no match for the three-headed space dragon. Ghidorah has Godzilla nearly dead, but Kong tackles him in time before he can kill him. Ghidorah blasts Kong, disorienting him and allowing himself to escape.

The dragon heals, and resumes his warpath, decimating major cities like New York, Beijing, Cairo, and more. Rodan and Mothra sense the destruction and awaken to fight the beast, chasing him to Tokyo. Kong assists them, but they are no match. Ghidorah grabs their throats using each of his heads, slowly suffocating them. Earth's doom is inevitable.

But before Ghidorah can crush their throats, he is hit by an atomic beam from behind, causing him to drop the monsters. It is Godzilla, having healed from his previous duel, but noticeably scarred. The four earthly monsters battle the destructive space dragon, destroying much of the city. Eventually, the dragon throws them off when the military attacks him, Ford Brody among them. He easily overpowers them, with Brody being the only survivor. Ghidorah prepares to finish him, but the monsters lunge from behind. Rodan pecks some of Ghidorah's eyes out while Mothra immobilises him by wrapping him in silk, and Kong rips his left and right heads off. Again, Ghidorah overwhelms the three kaiju, but is grabbed by Godzilla, who forcefully opens his mouth.

Godzilla charges a red spiral atomic beam and aims for Ghidorah's throat, who fires a gravity beam. The atomic ray pushes the gravity beam back, blowing Ghidorah's middle head off. The headless corpse slumps to the ground, the three heads rolling. Godzilla roars victoriously, before he and Kong exchange glances. The four monsters leave for their respective homes afterwards.