I'm almost of the mind that if A&M had a hand in it somehow, then it could be on the box set. But that's probably because I like the songs enough that I don't want to leave them off. Kidding aside, the associated labels are also a big part of A&M's history, but I don't how how (or if) they'd be represented on a retrospective set.
As for the tunes like "Raindrops" or the "Casino Royale" theme, there had to have been some sort of licensing deal to have the artist and same recording appear on two different labels. To use "Casino Royale" as the example--why would A&M want to give away rights to its most popular act at the time? And on the flip side, why would Colgems not want to keep a piece of the action and have the theme song appear exclusively on their own label? Remember that the Beatles' soundtrack to A Hard Day's Night appeared on United Artists, and from what I'd read, I don't think the film maker cared if the movie was a flop or not...they knew album sales would make them a windfall, getting a piece of the action that EMI and Capitol were cashing in on at the time.
Not only would you have some interesting licensing hurdles including distributed labels (Ode, Dark Horse, IRS, Windham Hill...well, okay, not IRS, which Uni owns now), but you'd have to start carving out more space on Discs 1, 2 and 3 to represent them.
As for "Casino Royale", by 1967, Herb and Jerry knew where the movie soundtrack market was...in the tank compared to five years before. As long as they had the rights to release the single and feature it on a TJB album, they knew they'd make way more money off the song than Colgems would. And Colgems had to take the deal. They weren't going to release the album without the theme song. If they cut Herb from the movie, then there'd be no conflict and A&M could release the 45 and LP anyway. It looks like there was more to the deal, and it all favored A&M...Burt Bacharach got to put "Bond Street" on the "Reach Out" LP and release it as a single...all on A&M (Dusty Springfield got to re-cut and release "The Look of Love" on Phillips).
"A Hard Day's Night" was a bit different. United Artists got the rights to the soundtrack album in the US only. It was released on Parlophone (EMI) in the UK. The US single was on Capitol, not United Artists. And it appeared on a Beatles LP on Apple ("Hey Jude", a compilation of singles not featured on albums) in 1970.