Mr & Mrs Lionel Barnett
Group From 21/10/1871 To 28/1/1960
We can find no information about "Mrs" but the Encyclopaedia Judaica lists a Lionel Barnett that could be our Lionel and Wikipedia adds some details: Orientalist. Born Liverpool. Joined the British Museum in 1899 and was keeper there of Oriental printed books and manuscripts 1908 – 36, and worked there again 1948 till his death. Also lecturer at UCL and SOAS.
We can find nothing to connect him with Hampstead or Waterloo or the House of Commons but his interests would not be inconsistent with the urge to preserve the fabric of old London. It's possible that the stonework was offered to the BM and when they rejected it he took it in charge.