If William was illegitimate you would expect his birth to be registered in his mothers maiden name
only if born in England - no civil registration on IoM (other than a very poorly used optional arrangement) until 1877/8 - illegitimates would actually be known under their father's name - the baptismal register merely states child of male and female with usually a note as to illegitimacy - there might be a presentment for fornication that would result in an appearance as a church consistorial court (records for such held by Manx Museum) but these records seldom if ever give the name of the child - it is possible tho rare for the woman to sue for maintenance from the father.
The Mormons with their insistance that the IoM followed the english custom re illegitimates lead many astray
on my Manxnotebook CD I have many of the records of this court - in court May 1861 a Elizabeth Bridson affiliated her child on James Peak who admitted the same
quick check of 1861 Castletown census indicates a James Peake s/o annuitant (gentleman crossed thru) James B Peake and wife Fanny - all children born England
More info is in the 1851 census when the Peake family is at College green Castletown James senior born London, with James W the son born in Shaldon Devon - judging from varied birth places for children family moved around and would appear to have only moved to IoM a little before the 1851 census
James junr would appear to be buried aged 34 at Malew 1 Sep 1874 (misread entry as James Peaks) - father James senr 5 Jun 1878 age 69 + mother Fanny 1887 age 74
at first I suspected Elizth to be dau of Richard Bridson + Cath Christian - bapt Santan 18301212 - Richard was a farmer at Ballfurt but the Richard gfa in 1861 is an Ag Lab living in Queen [Hythe] street Castletown (somewhat of a poor neighbourhood - see page on
www.manxnotebook.com under Castletown)- widower wife was Lucy Christie (she was illegitimate bapt Mal 18000400 )- however no Elizabeth in this family that I could see - Bridson is a common name and a Margt Bridson of Malew was also presented for fornication in 1860 and there is a gdau Louisa Leighall (a very non Manx name - Castletown was also home to the barracks and many illegitimate births registered to soldiers) - I wonder if Wm J [?james] Peak was actually a boarder rather a gson ?
yet more - the Peakes had moved to Castletown whilst one son was at King William's college - Robt Wm Youle Peake s/o J B Peake was entered as a Day boy in Aug 1848 and left mid summer 1851 noted as dying at sea c 1864 - it is possible other boys went to KWC but I only transcribed reg upto 1850 - KWC was a relatively cheap public school