I don't know what music played as the score you heard, but, for the excellent 2008 MGM Premiere Collection DVD release (which I own a copy of), that features the movie in beautiful restored form, two recent-years score options, each especially composed for this film, are included, one from 1997, the other from 1999. I can't remember which is the default score on that DVD (the other one didn't at all appeal to me, when I gave it a brief listen once), but can tell you that it suits and compliments the film wonderfully. I didn't watch last night's TCM airing of "The Lodger", but gather, from what you and the original poster have commented, that different music, entirely, played throughout it.
The music you're thinking of was not composed for "A Christmas Story." It's from Sergei Prokofiev's "Peter and the Wolf" (1936), a symphony for children that was well known from music classes and frequent radio broadcasts at the time that film is set. The French horns represent the wolf (and signal his appearance); the music's use in "A Christmas Story" is a deliberately humorous reference to the way Farkus is considered a predator in Ralphie's world. The only incongruity about it being used in the soundtrack for "The Lodger" is that it wasn't written until ten years after the latter film was made.