identifying the errant father is usually quite difficult unless the woman affiliates the child and gives some detail re the father's background (eg parish etc) - look at manx note book under illegitimacy - -there would appear to be no such affiliation that went through the ecclesiastical court (these records are only available to personal search at Manx Museum and I looked at 1830-1832 records - it may be (tho I suspect unlikely) that the mother sought maintenance money via the chancery court.
The child would always have been known by father's name - the entry in the parish reg might give additional details (rushen parish reg available on mormon microfilm) - if the father was well connected then probably no official record of any maintenance payments might exist - sometimes the child is recognised in the will

John is baptised in parish of his likely gparents with whom he is living in 1841, in 1851 shown as an apprentice blacksmith to an alexander davidson (nominally born in Rushen but I've not found bapt) - his probable mother married Thomas Leece in 1832 and had a family including an Alexander
In 1841 there is another child for whom I havn't found bapt living with Elizth Davidson (scottish by birth) - possibly presbyterian hence bapts not on IGI but I've looked at the non-confirmist bapts