According to this Wiki link, for which the source is apparently Many Years From Now, "I Saw Her Standing There" was "finished" at Forthlin Road in September of 1962. "Love Me Do" was recorded on September 4th and 11th. Awfully close.
The song was a Lennon & McCartney collaboration based on McCartney's original idea. Initially titled "Seventeen", the song was apparently conceived by McCartney whilst driving home from a Beatles concert in Southport, Merseyside and later completed at his Forthlin Road home in September 1962 with Lennon, while the two were playing truant from school.
Couple of things about this ...
1) Reliability of the wiki reference is called into question by the "playing truant from school" bit ... in 1962?
2) Assuming the rest of the reference is accurate (?) ... if memory serves correctly, "Love Me Do" was completed in or earlier than June 1962, when it was first recorded in Studio 2. "I Saw Her Standing There" being finished in the parlor in September 1962 means it was at least a few months behind in the song pipeline, if not more.
Then the question is, even if both songs were available to choose from for their 1st single in September 1962, which would have made the better first single
? Though probably everyone would say "I Saw Her Standing There" is a better song, maybe it's more important to release something with an original sound as the first single. The harmonica definitely made it stand out (best harmonica hit since Frank Ifield's "I Remember You"!). Also, though I'm not fond of Love Me Do nearly as much as other songs, I have to admit the harmonies there are much nicer and more developed-sounding than "I Saw Her Standing There". So maybe George Martin, as bright as he was, decided to market the essence of what he thought made the Beatles the Beatles - their originality and their vocal harmonies.
Another question that comes to mind is why didn't he at least use "She Was Just 17" as the 3rd
single, instead of "From Me to You" ? Maybe for the same reasons, sacrificing musical energy (which was WAY more on ISHST than FMTY) for vocal harmonies, and actually a great chord modulation (going to from the key of C to F for the middle eight). Or maybe he simply thought a loud rocker just wouldn't sell as a single back then ...
Just conjecturing, of course, and the only people around that would really know this would be Sir George, Paul, (Ringo?), Geoff Emerick ... I wonder if they lurk here?