What further information are you looking for? The Bethlem Hospital records which you have already found on Findmypast seem fairly comprehensive.
I wonder if perhaps you have confused two people with the same or similar name. The Charles Henry Shearman, whose Bethlem Hospital records are listed on Findmypast, was born in 1841 in Clapham (not in east London), the son of James Edward Shearman and Emily Shearman (nee Newton). Charles Henry Shearman's father, James Edward Shearman, and his elder brother (also James Edward Shearman) were both solicitors, living (in the 1881 census) at 77 Charlwood Street, St George Hanover Square. London (not in the Mile End/Shoreditch area.)
Charles Henry Shearman was admitted to Manor Asylum, Epsom, in 1899 and died there in 1901 aged 59. His admission record (including his photograph) can be found on the following site:
https://epsomcluster.com/patient-record/folio-38/