There were two Stynes farms in Ballyfoyle townland in Griffith's Valuation (28 Sep 1850) occupied by Patrick and William. This much earlier William was initially listed as occupier of two substantial houses, but in the valuation of tenements as altered on appeal by the sub-commissioners, one of his houses was listed as vacant, with William Stynes in the Immediate Lessor column. See links from
https://www.askaboutireland.ie/griffith-valuation/index.xml?action=doPlaceSearch&Submit.x=52&Submit.y=5&freetext=ballyfoyle&countyname=QUEEN%27S
Ballyfoyle is in Ballylinan registrar's district in Athy superintendent registrar's district.
RootsIreland claims to have indexed civil death records from Ballylinan for "1687-1901":
https://www.rootsireland.ie/ifhf/generic.php?filename=sources.tpl&selectedMenu=sources
This does not make sense since civil registration of deaths began only on 1 Jan 1864.
The RootsIreland index includes 8 Stynes deaths, ranging from 1873 to 1899, which can also be found on the less well-indexed but free civilrecords.irishgenealogy.ie. They include Patrick of Ballyfoyle in 1887 aged 81, presumably the Griffith occupier. They include only one William, who is of Ballyfoyle, but appears to have been only 38 when he died in 1883, which would make him too young to be the 1850 occupier. The age is blotchy, but both sets of indexers read it as 38:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1883/06344/4819600.pdf
James died in 1876 aged 57 (not in 1903 as estimated currently on WikiTree, possibly arising from confusion with a different James who died in a different registrar's district in the latter year):
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1876/020566/7216789.pdf
William's first (1889) marriage certificate confirms that his father James had died before 1889.
James's widow Eliza died in 1899 aged 74 and her death was registered by their son John:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1899/05796/4639671.pdf
The 1867 birth and baptism records of Alicia Stynes, daughter of James Stynes and Eliza Lalor, confirm that the parents lived in Ballyfoyle townland and in Arles Catholic parish.
There are two Stynes households in Ballyfoyle in both 1901 and 1911:
http://www.census.nationalarchives.ie/pages/1901/Queen_s_Co_/Killabban/Ballyfoyle/
and
http://www.census.nationalarchives.ie/pages/1911/Queen_s_Co_/Killabban/Ballyfoyle/
The heads of households appear to be two brothers (John and William) married to two Hyland sisters. The other Stynes/Hyland marriage is at
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1900/10337/5763398.pdf
However, the baptism that you are seeking for William is not to be found. As I have said, Ballyfoyle appears to be in the Roman Catholic parish of Arles, and RootsIreland claims to have indexed the relevant baptism records from 1821 to 1899. The National Library has various overlapping registers from 1821 to 1880, with all years covered by at least one register:
https://registers.nli.ie/parishes/0687
RootsIreland has no William Stynes in its extracts after 1833. Perhaps the name has been misread or is illegible or the priest forgot to record the baptism or the mother went back to her own home parish for the birth and baptism. The Stynes/Lawlor marriage is also absent from RootsIreland, another hint that the wife may have been from a parish for which early registers have not survived. However, nearby parishes seem to have good survival rates, although there may be gaps.
There are two James Stynes baptisms in Arles. Based on the Griffith occupiers and his age at death, I think that the Ballyfoyle James is far more likely to be the son of Patt Stines on 10 Nov 1828 than the son of James in 1835, as currently suggested here on WikiTree. The 1828 extract gives an address of Killabbin, presumably Killabban, which is the name both of the civil parish and of an adjoining townland. The 1835 record gives no address.
RootsIreland has an earlier (3 Aug 1836) church marriage record for a James Stynes in Arles parish, probably too early to be a previous marriage of Eliza La*lor's husband.
There is more useful information on the local geography at
https://www.genuki.org.uk/big/irl/LEX/Killabban
I will leave you to update all the relevant WikiTree profiles.