The answer is that Lady Balfour of Burleigh and Sir Robert Beecham, 5th Bt, are second cousins.
Lady Balfour's mother, née Jennifer Ellis MANASSEH (born 1930), was a twin with Jonathan Ellis MANASSEH (27 X 1930 - c.1 VIII 1984, this being the last date on which he was known to be alive) - their births were both registered in Kensington in Q4 1930, mother's maiden name Barnard. They appear to have had three older siblings, and it seems that their parents were Saleh David Ellis ("David") MANASSEH (c.1877 - 17 X 1962) and his wife (married 1917), Phyllis Annette BARNARD (1 VII 1896 - 28 III 1970).
David Manasseh (it seems that his initials and surname were usually rendered as E S Manasseh) was a son of Saleh MANASSEH (1840 - 1894) and his wife, Esther COHEN (24 VI 1848 - 13 VII 1924), who were also the parents of Emma MANASSEH (11 I 1887 - 2 IV 1972), who was married, firstly, to Edward CAIRN, formerly COHEN, and secondly, on 17 July 1930, to (as his second wife) the Hon Stratford Haliday Robert Louis TOLLEMACHE (25 VI 1864 - 30 VIII 1937).
Jennifer Mannaseh's first husband (married 1951, as his first wife) was John Edward Jocelyn BRITTAIN-CATLIN (21 XII 1927 - 29 III 1987), brother of the suo jure Life Peeress, Baroness Williams of Crosby, by whom she had three sons. Her second marriage, to (as his first wife) the 8th Lord Balfour of Burleigh, took place on 30 October 1971, and they were divorced in 1993.