Well, I have to disagree that "conspire" has any merit in the context of the song.
Here's a dictionary definition; "make secret plans jointly to commit an unlawful or harmful act".
What we have here, I believe, is the fact that everyone grows up hearing that song EVERY Christmas.
It has become part of our Christmas experience. We are comfortable with the lyrics. ALL the lyrics.
But, just imagine if that the song was written today as part of a songwriting class assignment.
Would we be surprised if the person grading the effort returned the lyrics and had drew attention to the fact that the word "conspire" is not appropriate in the context of the rest of the song.

Is it possible that the lyricist "settled" on a poor choice of word.
The couple in the song are making plans while they sit by the fire. But, of course, the word plans appears in the next line and, of course, wouldn't rhyme with fire.
No, I think "conspire" was a very poor choice, and I bet the lyricist wasn't happy with it neither!