Dora Quayle was an unusual name. There are only 5 in the 1911 census, including the one you mentioned previously: QUAYLE, Dora Mylechreest, aged 12 Isle of Man.
Adoption in England and Wales was a private arrangement before the Adoption of Children Act, 1926. Registers were started in 1927, and before that there is no public record of adoptions. If your mother's birth cert was in her birth name, how did she make the connection between this and the name she had grown up using? Someone must have given her information, because this is the usual sticking point when looking for people adopted pre-1927, unless the adoption was registered later (which a lot were). If her information came from the Catholic Church, it is possible that they have more details. I think you need to follow the same trail back that she herself travelled to find her birth cert., and find whether she was formally adopted or fostered. If fostered the Church would know whether anyone made payments towards her upkeep. Were any solicitors involved? Was she mentioned in her father's will? The old Somerset House will indexes are much faster to search than deaths, and you could check Darwall as well as Darwell. There is a Darwell death on FreeBMD with "S" as a second initial which could be worth checking for a will. The post-1858 wills and indexes were moved from Somerset House to the Family Records Centre, and I think the indexes were copied to fiche and are (or were) available in several major libraries.
Another thought about his surname is whether the birth cert you have is a handwritten or typed copy from the register, rather than a photocopy of the actual entry? If not a photocopy try asking for one from the local RO which has the original entry. It could be a good idea to get this anyway, because the GRO is notorious for the number of errors made in their copies.