I’ve spent a little time working on this one today, and while I have no definite answer, I’ve at least come up with a theory.
First of all, what is that other song? Well, it is indeed called “She’s So Fine,” and it’s from Jimi Hendrix’ Axis: Bold as Love album. You can hear the song at this YouTube link. The Wikipedia entry for Axis: Bold as Love devotes a single sentence to “She’s So Fine,” but it is enough to tell us a couple of important facts about the song:
So this “She’s So Fine” was composed and sung by Redding, and was included on Hendrix’ Axis: Bold as Love album.
Those are the facts, and now we get into some (hopefully not too shaky) theory. Here, we need to look at two things: the efforts of well-meaning but misinformed Internet posters, and Flash Cadillac’s reputation as a cover band.
Thinking back to those pre-RIAA-lawsuit days of about ten years ago when file-sharing was popular, it was surprising to see all the errors in the listings of songs available. There were spelling errors, of course (Led Zeplin, Lead Zepelin, Led Zeppllin, etc.), but there were also many errors in songs, artists, and composers. Some were obvious errors (The Who never did “Rocky Mountain High” for example), but others weren’t so obvious.
Let’s take an example. If you were asked, “Who recorded ‘The Loco-Motion’?” your answer would likely give away your age. Older Boomers might respond with “Little Eva”; younger Boomers with “Grand Funk Railroad”; and Gen-X with “Kylie Minogue.” All correct answers, by the way. But what would you do if the person asking said, “No, it was Carole King”? Because this is a valid answer too–Carole King (along with Gerry Goffin) originally wrote “The Loco-Motion,” and recorded and released it in 1980, years after she wrote it, on her Pearls album. So here we have a composer, who has written a song, that is better known by other artists, who has recorded and released it anyway, and it’s one of a number of versions floating around. But given that King’s version never did as well popularity-wise as Little Eva’s, Grand Funk’s, or Kylie Minogue’s, “Carole King,” while a valid answer to the original question, is going to be an unusual answer to the question.
Now, let’s assume that the one person who really and truly believes that Carole King did the one and only version of “The Loco-Motion” worth listening to, puts the MP3 of the King version out there. Somebody else sees it, and hearing a woman’s voice doing it, and knowing that Little Eva did it originally, corrects the artist to Little Eva. Somebody else, who is a lot younger, listens and corrects it to Kylie Minogue. Someone else doesn’t listen, but knows it’s a Grand Funk song, so they correct it too. Pretty soon, King’s version is out there and widespread, but is no longer attributed to King. You simply cannot find King’s version–at least, not using her name.
Hold that thought a moment, and we’ll look at Flash Cadillac’s reputation as a cover band.
We already established that Flash and his boys aren’t always a cover band–they did, after all, originally compose their version of “She’s So Fine.” But most importantly, they are seen as a cover band. I believe this is mentioned in the OP.
But think, for a moment, of Sha-Na-Na. Fifties greasers doing old songs. Another cover band, right? Not quite. Among their early recordings are a number of original compositions, not all of which follow the “greaser doo-wop” model. They are nearly impossible to find nowadays (unless I dig my boxes of old LPs out of the basement, I suppose), but Sha-Na-Na albums like Live at Columbia University and The Night is Still Young, both of which are from the early 70s, contain a number of original compositions as well as covers. But I haven’t been able to find much on Sha-Na-Na’s originals out there. Why? Because Sha-Na-Na is a cover band, isn’t it? 
Now, let’s put those two thoughts together. Flash Cadillac is, and always has been, a cover band. Everybody knows this. Everybody also knows that “She’s So Fine” is a song on rock legend Jimi Hendrix’ Axis: Bold as Love album. So when a well-meaning but misinformed person sees an entry for “She’s So Fine” attributed to Flash, but it’s going on about “bobby socks and a ponytail,” well, whoever posted that must have made a mistake. They must have meant Redding’s version on the Hendrix album, because Flash Cadillac only covers songs–that band doesn’t do originals–and so it must have covered this Redding song. This must be one of their covers. Let’s just fix that up…
And just like you’ll never find Carole King’s version of “The Loco-Motion,” due to the actions of well-meaning but misinformed people, you soon won’t ever find Flash Cadillac’s “She’s So Fine.” See what might have happened?
That’s my theory anyway. I hope it makes sense and that you find it useful. I sure did; it helped give me a break from some boring work.