Chittick, sound very posh.
Checked the Tithe Valuations 1823-1837 and no record, they might not be available as this is the National Archive website.www.genealogy.nationalarchive.ie
Next Census substitute too late for your David but there
are 27 surname Chittick's in Ireland, and the following Parish's and Poor Law Union in 1862.
Parish Drumkeeran, PLU Irvinestown
Parish Enniskillen, PLU Enniskillen
Killesher, PLU Enniskillen
Kinawley, PLU Enniskillen
Rossary, PLU Enniskillen
Magheeracross, PLU Enniskillen
Dromore, PLU Omagh
Drumragh, PLU Omagh
A Poor Law Union covered an area of 12 mi. within that
all parish's, townlands, villages and towns are under that authority.
David was not David's father as there are none listed unless he was deceased by that time (1862). This is one of the lastest records for the Griffith Valuation that went from 1846-1864 in counties.
There doesn't appear to be any of the above parish's in the Fermanagh County Church of Ireland surviving records (assuming are Protestant)and only Drumragh has records back to 1801, next surviving parish records are 1836.
There are several William's in the list one could be the older son who took over the lease of the property? only one appears as IN FEE which means owned by.
James Chittick is the landlord of 152 acres and has several tenants in the parish of Drumkeeran, Townland Drumboarty, Poor Law Union of Irvinestown.
#8 James has a bog INN FEE 5.0.0 & #9 2.2.17 in the townland
#3 William is renting bog from James 3 acres 0 roods and
No home for James so he must live elsewhere.
James in Townland Windmill Hill, tenant of Earl of Enniskillen
#81 Henry Street land 4.1.25
Parish Enniskillen, Union of Enniskillen Townland Coleshill
#20 James land 3.3.1
Isabella Chittick, Parish Enniskillen, Earl of Enniskillen
#9 Anne-Street, house, outbuildings,yard and small gardewww.askaboutireland.ie
(for the others)
There are James x 5, John and George who were on the Electoral registry 1832, 1837 & 1838 (you had to own land to be able to vote)
4 of the James are in Enniskillen, the others just say
Father William is a possibility if the 1st. son of David is named William, 2nd. son her father
(even though the Irish naming pattern is mostly thought of as an RC thing it applied to CofI as well in most cases at least in earlier years)
There is 1 Chittick in the Fermanagh British telephone book, address comes with a postal code. In case he is very cold I would write a letter immediately.
For free genealogy in Northern Ireland try googling:
Irish Genealogy Toolkit