AFAIK, Liberty had no kind of stereo or binaural facilities at that time; borne out by Martin Denny's remarks that he and his group had to record "Exotica" all over again in stereo after-the-fact to fulfill dealers' demands for a stereo version that did not exist.
There's a very good reason for that. The monaural
Exotica was recorded in a RADIO STUDIO and in the Fall of `56. The stereo version wasn't recorded until almost a year later, by which time twin-track had begun to be installed at a number of other studios - hence the 1959 remake of the Chipmunk Song being in stereo - and the original binaural or half-inch series of work parts being located, re-synched and remixed.
By 1956, major labels on the West Coast (RCA, Columbia et al) weren't interested in stereo or binaural, even though everybody had at least one back in Midtown (Manhattan) - ostensibly to capture the very best of classical music sessions.
Therefore it fell to the next-in-line studios. Reports differ as to whether the first twin track in Hollywood was installed at Capitol Studios upon its' completion in 1956 or at Decca Records on Melrose upon its' remodeling that same year.
In 1957, United Recorders (later United/Western Recorders, famous for its' Beach Boys sessions) was a prime example, where owner-engineer Bill Putnam saw the writing on the wall and installed a twin-track recorder and a binaural mixing board with his own money. Shortly before or afterward, owner-engineer Thorne Nogar installed one at Radio Recorders.
Stereo/binaural sessions at both Capitol and Decca were rare until at least 1958, by which time twin-track machines had begun to be installed all over Hollywood. But by Summer, 1957 since both Nogar and Putnam were putting money out of their own pocket for not only the twin-track deck itself but also for the mixdown sessions necessary for the production of a mono record - both men subsequently had stockpiled huge libraries of twin track tape from the most lucrative acts of the late `50's.
By the early 60's when stereo began to take off, both men scored lucrative deals with the major labels for re-issuing what amounted to twin-track work parts and the cost of producing them - which financed yet ANOTHER renovation in both studios in the Summer of 1962.
It isn't necessary to involve a whole Hollywood studio to record on film; the L.A. area certainly had many film labs and editing shops set up to service the literally hundreds of small production companies making very low-budget TV shows, industrial and educational films, cartoons, and commercials.
But again - by the late 50's - especially for the small and/or field productions - the use of Nagra (pilot-sync quarter-inch reel-to-reel decks - either single track and sync on 2 or twin-track and sync in center).
It's possible that the original elements were recorded on tape, "dubbed" to film by one of these companies, then synched and mixed on film.
I still doubt that - Les Paul's involvement notwithstanding - that the session was recorded - or dubbed - to film - mag or otherwise - simply because if you HAD to transfer to film - you'd need sync.
On a twin-track recorder, you only have one track if you need the other for sync. (Center-track sync was still a few years away yet.) Therefore you'd be back to flying elements across to that recorder and/or adding in subsequent elements live over the top of it, flying it back over here to this recorder - doing the same - and over and over and over
Although if the story of Paul's involvement IS true - then that would make sense as well because Capitol Records - with whom Les Paul had been affiliated before their acquisition of Liberty and United Artists Records - could have had the session moved to Capitol Studio A where their first twin-track had been installed in 1956 - used ``as an afterthought'' without any specially dedicated stereo engineer or anything - to cover a Nat King Cole album - owing to the fact that the first two songs of the first session (of four songs in a three hour session) never made it to the twin-track due to a patching problem which was corrected for the last two songs of the first session and the four subsequent sessions.
StereoScout over on the Stereo Sync board
http://bsnpubs.websitetoolbox.com/?forum=6491 would have more in-depth research at his fingertips regarding both early stereo in Hollywood as well as session tracking information.
. Very infrequently I'll come across a 16rpm single or Lp, but I no longer have (nor want) the capacity to play that speed, so they mean nothing to me beyond their status as curios.
