Godzilla franchise and King Ghidorah belongs to Toho productions.
Gate: thus the JSDF fought there anime belongs to A-1 Pictures.
I had this idea since I watched the Gate anime where in one episode they mentioned about the Flame Dragon being a one-headed King ghidorah. This is an idea for others to use since no one has yet written something like this and I'm terrible at writing stories.
The story takes place when Ghidorah arrives to Falmart a few years before the opening of the gate and the arrival of the JSDF. He arrived at Falmart being contained inside a meteor (The same way on how he arrived on earth in the movie, Ghidorah: The three headed monster). The Empire would notice the meteor landing and sent soldiers to investigate the meteor. When they reaced the meteor, they discovered that it was attracting their metallic armor and weapons to it (the power of the magnetic field around the meteor fluctuates in strength from day to day; in one day, the field would be strong but on another day it would be weak. This is taken from Ghidorah's first film appearance where the meteor that he is inside of generates a magnetic field that attracts any metallic object.) After a couple of days, Ghidorah emerges from the asteroid as a fireball before taking shape into his golden three headed dragon appearance (His appearance in Ghidorah, The three headed monster). After emerging from the asteroid, he would then start attacking the empire and the surrounding villages.
After a few years, the gate would open and the JSDF will arrive on the special region. The story will diverge from canon because of Ghidorah's arrival a few years back. And how will the Japan and the Empire deal with Ghidorah and how will the public react when they see Ghidorah?
A Few things to note when you try to write this story:
1. In the anime, the Godzilla films exist in canon because of how it was mentioned that the Flame Dragon is a one headed King Ghidorah. The world alongside Japan have either seen it or heard about it and they are probably not expecting for Ghidorah to exist in the Special Region. The kaiju in Godzilla films don't exist on Earth since they are works of fiction. Since the Godzilla films exist in canon, the film company Toho Productions would also exist. How would they react when one of their most iconic kaiju exists in the Special Region?
2.You can choose whether for the Flame Dragon to fight Ghidorah or not. Ghidorah could also replace the Flame Dragon before the JSDF arrive and be seen by the 3rd Recon team destroying the forest anlongside the elf village where Tuka lives.
3. You can use any monster from the Godzilla films besides Ghidorah to exist in Falmart.
For information about Ghidorah and other Kaiju, use the Wikizilla and the Gojipedia wiki.