I couldn't find anything definitive in Google books, but Pitney's record was definitely on the market by March 31 1962, when it was mentioned in Billboard, and it peaked on the charts sometime that spring or summer (Wikipedia says in April).
It's possible that the Fairmount Singers version was released first, because of the business relationships between Bacharach, Paramount Pictures, and Dot Records. Bacharach had been working for Paramount's Famous Music publishing arm since the mid/late 1950s, and the song was published by Famous. Dot Records was owned at that point by Paramount, so Dot artists might have had "dibs" on compositions controlled by Famous Music. There's the question of who really owned the song and could approve a first commercial release. I don't know if Bacharach and David had that power, or if Famous Music did due to the nature of the commission they made for the song to be written.
In some cases in those days, studios got very favorable terms on the copyright split (sometimes the studios owned the song outright) for songs commissioned for movies and television, but I don't know the details for any of Bacharach's early Hollywood work. Talking about a different song commissioned by Paramount, Bacharach told Serene Dominic
We were doing songs like movie promotional songs for hire. It’d be $500 a song. “Desperate Hours,” I remember that.
"For hire" is often used to mean the work becomes the property of the firm that commissioned it, but I don't know if that's what Bacharach meant. It could just mean he got a $500 advance for writing the song and still retained some share of the copyright royalties.
In any case, it seems a little academic to talk about the Fairmount version as "the original" based on the release date alone. Bacharach in 1962-1963 had a publishing deal with Pitney's manager Aaron Schroeder, and was producing his own songs for Schroeder's Musicor label artists, including Pitney. Pitney's version was the version that Bacharach thought was going to be in the Paramount movie (though as often happened in those years, the studio decided not to use it). The arrangement on the Fairmount Singers version is pretty bland and generic, and doesn't feel much like Bacharach had anything to do with it.
On a side note, Pitney told journalist Bob Greene that he was literally in the middle of the recording session, thinking the song was for an unreleased movie, when someone at the studio mentioned to him that the film was playing at a theater down the street. Wikipedia says the film opened April 22, 1962, and Billboard says the song was in stores in March, so that Pitney story seems possibly to be simply a great showbiz tale. The Fairmount recording has the "inspired by" line on the label which indicates that Paramount had already passed on the song when the Fairmount recording was made. If the Feb. date for the Fairmount single is correct, that makes Pitney's version of the story that much harder to buy (assuming the film really didn't hit any screen in the U.S. until April, though in those days, both films and records often had "rolling releases" where they'd show up in one part of the country before other regions had access to them).