I've brainstormed ideas for some other Toho monsters. It takes some work to make their origin stories fit the realistic tone of the Legendary verse, but it's possible for most of them. King Ghidorah is a real stretch, though.
1) On Saturn's moon Titan, there are micro-organisms that feed off the abundant methane deposits; sometimes they agglomerate into a gigantic amorphous mass. Millions of years ago a meteorite threw off a section of Titan's surface into interplanetary space, carrying a colony of these micro-organisms. The dormant microbes rode an asteroid into the inner Solar System, where they eventually fell into Earth's gravity well.
Between the events of Joe Brody's terrible day at Janjira and the MUTO incident in San Francisco, an oil rig in the Gulf of Mexico got a new pump jack with which to dig into a seafloor oil deposit. The drill disturbed the resting site of this mysterious hydrocarbon-consuming organism. It surfaced and caused catastrophic damage to the oil rig. When the creature was tracked by Navy vessels and placed under surveillance, MONARCH gave it the codename "HEDORAH".
2) When Joseph Stalin ordered the expansion of the gulag system into northeastern Siberia, his prospectors encountered an enormous rhinoceros-like creature sleeping on the Central Siberian Plateau. Though docile at first, the repeated encroachments by gulag construction teams agitated the beast until it went on a rampage, killing many hundreds of laborers and a few Communist party officers. The creature displayed no interest in eating the humans it killed, but it was observed using its nose horn to break chunks of rock off of mountain sides and eating the hard bits that fell out. The gulag camps were repurposed into providing ores to feed the creature and keep it from going on another rampage.
When Khruschev became the leader of the USSR, he ordered the creation of D.R.E.K.A., a clandestine scientific-military unit whose purpose was to study the creature and if possible weaponize it.
Reports of an anomalous creature in Siberia reached Washington after the CIA ordered an overflight of the region. M.O.N.A.R.C.H. determined that the creature was a "lithitroph", feeding on mineral matter, as opposed to the radiotrophs that were the MUTOs and Godzilla.
They named the beast "BARAGON".