Other than the Foundation website you mention, the National Library has RC records these were microfilmed by the LDS.www.familysearch.org
- older records -look for the link to the previous website it is on the right. The National Library collection is not complete nor is the LDS but the County website should of picked up from the parish records any records that were available as they collect from many sources.
There are several references to Philip Cleary's and Mary Heffernan's but doesn't look like there are records for yours.
eg: Philip Cleary 1785 Diocese Cashel & Emly married 1811 Jane Tompson
RC-The Parish of Carrick-on-Sir 1784-1880 marriages 1788
RC-The Parish of Cahir 1776-1793 and 1809-1864 m: from 1776
C of I St. Nicholas Main St. 1845-1864
this maybe why some records seem to be missing?
There are 19 Cleary's in the online white pages for Tipperary and 74 Heffernan's - click on residential and look for places in the area you are looking for. Then you can write some letters. Especially look for Philip as it is obviously a name passed down through the family.
Ancestry recently put the Tithe records for the 1820's online, just names and place (no land descriptions as for Griffith Valuation)This will give you the previous generation names. The are the odd records for the Tithe's but the complete collection is at the National Archives but only searchable in person.
there maybe some gravestone inscription transcribed by a Heritage Society or try a google of Cahir.
The Tipperary Library has some 'local studies' that include some gravestone inscriptions www.tipperarylibraries.ie/local_studies/index
Can't think of anything else?