There are still Killoran's and Casey's in the Sligo telephone directory.
You should be able although your dates are super early to cross-reference the two families in Sligo. Using Griffith Valuation on www.failteromhat.com
look for common Parish's for the two names. A lot of work as there are 77 Killoran's and 111 Casey's.
Parish records for Sligo: About 26 parish's for each RC/CofI
The earliest Parish for RC is 1762, no more till 1805, then 1817+
Church of Ireland 1 1796, then 1803, 1815, 1825+
The odds of having the right parish are not good.
A Parish birth record gives mother, father and mother's maiden name, parish/place/date
Mark is an unusual name for the time so you might be able to find a birth record using just his name and birth year. Its worth a try you will either get one or not. You can find the above mentioned Parish records on a Irish History Foundation to which Sligo belongs. Its a not for profit but fee paying website. Is about the only way to find records without visiting Ireland. If you can visit there are parish records kept at the National Library, some of which were microfilmed by the Latter Day Saints, but you would need the Parish to have the microfilm #. going through thousands of names would take forever.
Church of Ireland records are easier as all surviving Parish registers are kept in one place the Representative
Church Library in Dublin.
This is the Westmeath board so you should post on the Sligo board as well.