Slashfilm provides a good breakdown of the ending of Kong: Skull Island, which is corroborated at Refinery 29.
Special thanks to Oliver_C and BCdotWEB for providing these links, and thus providing the key to the answer I was looking for, as I did not think to remain until the credits ended!
Kong: Skull Island is heavily referential, most notably to Apocalypse Now and Moby Dick, although there may well be references I didn't catch. The inclusion of this specific song is likely not random, as it constitutes one of the most famous scenes in cinema: Ending of Dr. Strangelove.
The support for this idea comes via the Kong sequel teasers:
"Kong: Skull Island isn’t just a new take on the classic movie monster that originated on the big screen back in 1933. It’s also the second film in the new cinematic universe that launched with Gareth Edwards’ new take on Godzilla in 2014. Though Kong: Skull Island takes place all the way back in 1973, it prominently features the government organization Monarch, the same department that kept the existence of Godzilla and other monsters under wraps in Edwards’ film."
"They’ve found evidence of other monsters existing. A film reel begins to play, and while it doesn’t show any actual pictures of these monsters, there are photos of cave paintings that reveal not just Godzilla, but also the silhouettes of other monsters."
"The final image shows a painting of Godzilla fighting with a three-headed monster and the scene cuts to black…and we hear the iconic, shrieking roar of Godzilla."
"This post-movie scene merely confirmed what fans of big monster action movies have long hoped for: Godzilla's returning in 2019 for Godzilla King of Monsters."
"Even more importantly, the teaser officially confirms that King Kong and Godzilla exist in the same reality. By linking Kong and Godzilla, Legendary Pictures has given birth to a whole new universe of action movies. By 2020, the monsters will duke it out in their very own movie, Godzilla vs. Kong."
Nukes were mentioned several times in Kong: Skull Island, including the implication that the US had tried to nuke the island in decades past, and may well attempt it again.
It is highly likely the inclusion of that specific song at that specific point in the film is a reference to Dr. Strangelove because:
Godzilla is an allegory on the dangers of nuclear weapons, which is also the main theme of Dr. Strangelove.