The original idea for this crossover comes from the hand of series writer Shinichi Sekizawa, who submitted a manuscript of the proposal on November of 1965. Sekizawa's concept featured several characters from the Batman universe, including Robin and Commissioner Gordon. To actually battle the King of the Monsters, Batman and his sidekicks would have utilized several vehicles to engage in combat, including the Batmobile, the Batcopter and the Batcycle. Interesting to note, the concept also featured the introduction of a weather control apparatus, an idea which would later be worked into the Son of Godzilla script, and another device to control Godzilla himself, which would possibly indicate another antagonist being behind pitting the two characters against each other.
In terms of Sekizawa's motivation, his reasoning seems clear in these drafts: to try and repeat the mammoth international success that the King Kong vs. Godzilla crossover enjoyed. This is made apparant in allusions to the earlier 1962 film, as even this extremely early take at the concept included mention of Godzilla's battle with King Kong, which was noted as being included with stock footage of the two titans fighting.
However, the concept never got far enough for a full fledged script to have been created. There are many questions related to the proposal still unknown as well, such as if DC comics was ever actually approached with the idea or not. It's also not known to what degree the then recent Batman TV series, which debuted two months after Sekizawa proposed this idea, might have played, if any. It's also interesting to note that references to "Batgirl" are made in the concept, which would predate the Barbara Gordon version of the character that would later become synopsis with the name in 1966.
The Dark Knight, with the help of Robin, does battle with the nuclear menace, Godzilla.