I believe the source of this belief "both Lp's were recorded at the same time" comes from a lengthy Mick Jagger interview in Rolling Stone a few years ago. Jann Wenner conducted the interview and basically takes Mick through the years from the very beginning asking about the various recordings/events in the band's history.
Jann asked about a particular song (Midnight Rambler, I believe) and Mick responded "What album was that on?" and explained that "Beggar's Banquet" and "Let It Bleed" were recorded together or at the same time or something like that.
They were indeed recorded at the same time if "at the same time" means spring 1968 through fall 1969.
Mick Jagger, never known as a meticulous note keeper, can be forgiven his vague recollections about chronology given the intense situations among the band's personnel (Brian's death, Mick Taylor's coming onboard), performance pressures (NME May 1968, R&R Circus December 1968, Hyde Park July 1969, plans for US tour 1969) as well as incredible productivity in the studio which not only produced two of their finest albums (BB and LI
but also two of their finest singles (JJF & HTW), Jagger's solo single (Memo From Turner), songs for future albums (Sister Morphine ended up on Sticky Fingers), material which ended up on Metamorphosis and Jamming With Edward, and many out-takes which are readily available on assorted bootlegs-all occuring within an 18 month time frame!
Moreover, from the very beginning of the Stones recording career, studio chronology never equaled release chronology, with songs ending up on different LP's in different countries at different times. Even after the departure of Andrew Loog Oldham, this remained the case. "Loving Cup", for example, was introduced by Mick at the Hyde Park concert as being "from our new album" (the as yet unreleased "Let It Bleed") but it did not surface on an LP, in a different version, until 3 years later, on "Exile on Main St."