This is my own speculation and is not based off of any spoilers or knowledge of the actual plot in Godzilla Vs King Kong. All rights are reserved for the copyright holders of the respective works and no attempt is being made to profit off these concepts in any way.
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MONARCH is tracking a large and aggressive wolf-like beast. They keep losing track of it, despite it's 100ft+ size...and it appears to be searching for something. It crosses the mainland in a few places, leaving very strange tracks. Huge, but not very deep.
The humans finally discover it has snuck onto Skull Island in what is revealed as a quest to absorb more of the weird bugs it is made of. It leaves weird footprints because it isn't nearly as heavy as they had believed it was due to it's composition.
On the island, Monarch keeps its distance to avoid infuriating Kong. Due to this, they are unable to get a clear idea of what is going on through the mist covering the island this time of year.
The Beast, which we see sucking up a couple large swarms of the insects, tries to ambush King Kong. During the fight, the humans attempt to collect some bug samples. After losing due to Kong's intelligent use of a tree as a gigantic fly-swatter, The Beast flees the island to another area where the bugs it is made of live...running directly into Godzilla's path.
MONARCH has trouble keeping the bugs for study due to their hardiness and ability to generate heat. The team pursues The Beast, while still trying not to antagonize Kong and now Godzilla, who are both hunting the same creature. MONARCH finally gets a good look at Kong's colossal size and realizes he wasn't fully grown at their last encounter. He is now at least 75% the size of Godzilla.
The aggressive Beast attacks a surprised Godzilla who rips into it with tooth and claw. The bug composition of it's body leaves it mostly unharmed from bites and tail swipes. We see more swarms joining this one, making it noticeably heavier and able to bite back.
Finally Godzilla has had enough and lets loose with his breath weapon. We now discover the bugs are especially vulnerable to atomic blasts as it loses 1/3 of it's mass, quickly regenerating the lost limbs. It is fast enough to almost entirely dodge additional breath attack leveled at it, but it still retreats from a quickly outdistanced Godzilla.
MONARCH is monitoring the encounter and organizing a rescue/evacuation of a small town nearby that works a huge rock quarry.
A few hundred refugees in a caravan set out to escape the danger, while most citizens assemble for evacuation as the military ordered. Heavy helicopters are brought in to clear out the people who stayed, while the caravan reaches a destroyed road that traps them on the mountainside. MONARCH decides to save them.
Kong arrives, clearly more agile than Godzilla and at least as strong. He catches the Beast by surprise and unknowingly slows it enough that Godzilla catches up as well.
Both huge territorial Kaiju face off with their mutual foe trapped between them. They end up quickly crushing and pulling the smaller creature in half and proceed to become aggressive towards each other, pissed the other is involved in what each considers their personal fight.
MONARCH considers intervening due to both monsters being relatively friendly towards humans, afraid of the nightmare scenario of one of humanity's two protectors killing the other. Due to politics, most of their efforts go towards rescuing the trapped caravan who are fleeing on foot now.
Kong takes a glancing hit to the body from atomic breath and, though hurt and singed, is able to withstand it by quickly putting hills and mountains between him and Godzilla. (MUTOs took direct hits multiple times until it was aimed inside one, so Kong should be able to handle it in short bursts). Kong realizes humans are nearby (Godzilla doesn't see them due to the mountain) and he rips the side of a granite mountain out of the ground and proceeds to block the next breath attack. This saves the entire group from instant death.
Kong's actions allow MONARCH to quickly rescue the trapped humans. Godzilla acts surprised to see humans there and stops his breath attack...Kong then uses the distraction to hit Godzilla with his huge rock until it crumbles from hitting Godzilla's face.
A side scene depicts MONARCH employees arguing over a code name for The Beast. It is discovered the bugs are a colony organism and are smarter the more of them there are, though there appears to be an upper limit. They are immune to heat, but seem vulnerable to the particular type of radiation Godzilla's breath is made of. The heat they generate also grows with their numbers and is capable of causing atomic reactions in oxygen. Effectively if enough of them work together, they could ignite the atmosphere in a massive fusion explosion.
A clearly bruised/bloody and unphased Godzilla uses Kong's surprise at the rock shattering to spin and deliver a tail strike that knocks Kong completely off his feet. As Godzilla stands over a dazed and slow-to-recover Kong, the Beast, now reassembled in a slightly larger new form ambushes Godzilla with a massive overbearing attack. The Beast's new form reveals it as Destroyah. MONARCH confirms they decided on Destroyah as it's code name.
The lead scientist notifies command that a huge colony of the bugs appears to be native to the area, meaning a 2nd Destroyah could arise. It quickly becomes apparent to the humans that the Kaiju are fighting the new Destroyah now, not the original. The original Destroyah, smaller than the new one, crushes a tank brigade that was moving into position and moves back towards the fight.
The humans now safe and their military assets under attack, MONARCH decides to intervene. They cannot allow humanity's two defenders to die against this oxygen-destroying creature! Scientists speculate Destroyah cannot control it's form to move while generating it's oxygen destroying attack and communicates this to the military.
Kong recovers from his daze and tears into his rapidly regenerating true enemy, accidentally freeing the wounded Godzilla. The human military moves into position and jets are scrambled. The original Destroyah also moves into view, smaller than before due to battle damage, and merges with the new Destroyah. The two Kaiju Kings stand side by side against the now much more powerful Destroyah.
The new Destroyah is faster, stronger and larger than previously seen, even larger than Godzilla. It dodges breath attacks and thrown boulders. The jets sent in by MONARCH cause Destroyah to reveal it's breath weapon, obliterating most of the strike force. It then turns its burning breath on it's larger enemies and manages to hurt both of them. They split up and it is forced to stop using it's breath to fight.
For the rest of the fight one or two surviving military pilots lob missiles into Destroyah whenever it tries to stand still in an attempt to keep it's breath weapon out of the fight. This is partly successful in reducing how powerful a hit Destroyah can deliver.
Other than the pilot sneak attacks, the only time Destroyah takes any damage is when our favorite Kaiju attack at the same time, making it tougher to dodge both attacks. The fight ends up happening through the small town that is quickly obliterated. The survivors cry out, watching drone images from the aircraft carrier they were taken to.
Finally Destroyah tries to use it's breath weapon again and a boulder Kong throws is a direct hit. This pins most of Destroyah in the rock quarry and, before it can disassemble itself to escape, Kong repeatedly jumps up and down to crush Destroyah. When he sees Godzilla, he leaps out of the way and Godzilla burns it apart with his atomic breath. Some of the bugs are seen sneaking away, too small for the Kaiju to easily see.
The humans are humbled by their inability to help in the face of such awesome power, but glad their side won. MONARCH begins assembling a team to hunt the surviving bugs before they grow numerous enough to be a threat again. Politicians debate if this is truly a win, while citizens begin demanding action to protect them from all of the monsters.
The two Great Sovereigns share a long look of mutual respect, then Kong turns and returns to Skull Island. Godzilla seems to take note of where Kong went, then goes back to his interrupted hunt for enemy Kaiju.
A final scene shows plans for a huge robot that would protect humans from the rampaging Titans. Mecha Godzilla.