My vote for the most convincing fictional song is ‘That Thing You Do!,’ from the 1996 film of the same name. This track is the one-hit wonder by 60s pop band The Wonders, and it perfectly captures the sound and spirit of that time.
That Thing You Do! (the movie) was the writing and directing debut of Tom Hanks. It’s a charming film that never quite found an audience. It chronicles the short career of a band called The Wonders (renamed from The Oneders after everybody mispronounced it as the O-need-ers) that sends this song up the charts before falling apart.