I didn't like a few things about this couple's possibilities:
Here is a possibility to look at:
Peter TRAINOR Kilbroney husband of Elena Campbell ; father of Alice bpt. 31 Jan 1808 & Arthur bpt. 25 Apr 1811 at Kilbroney Catholic Church PR [= Parish or Church Register]
found at
http://rosdavies.com/SURNAMES/T/TrainorTreanor.htm
1. Ann (Trainer) McLinden didn't name any of her children Alice or Arthur, at least not so far as you have found.
2. I didn't find any of the 4 McLinden children, Peter, Ellen, Anne, or Mary in the free index of baptism records at findmypast.ie, although the Kilbroney Parish records are available for that time period.
3. I didn't find the marriage of Ann Trainer and Bernard Mclinden there either.
So here's another theory:
Ann and her husband Bernard McLinden were having children from 1826 to 1840, so Bernard was alive when the Tithe Applotment Books for Down were taken, from 1823 - 1837 according to
www.titheapplotmentbooks.nationalarchives.ie.
I ran a search for Bern* Mc*lind* in the Tithe Applotment Books and only found:
clonduff parish: McAlindon, Bernard Townland: Cleomack
clonduff parish: McAlindon, Bernard Townland: Island Moyal
clonduff parish: McAlindon, Bernard Townland: Leyod
There are plenty of Trainors and Campbells in Clonduff as well.
Parish records for Clonduff/Clonuff do not start until 1850.
(see
https://www.johngrenham.com/browse/counties/rcmaps/downrc.php#maps/ and click on Clonuff.)
Which would explain why the children's baptisms aren't showing up.
This assumes that findmypast.ie's indexes are accurate and complete for parishes in Down that are available. Also that Bernard McLinden met the qualifications for the Tithe Applotment Books, but it looks like all you had to do was occupy 1 acre of land.
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There are a lot of McLindens in Clonduff on Griffith's Valuation (1864), including a Bernard:
McAlindan, Arthur
McAlindan, Daniel
McAlindan, Denis
McAlindan, James
McAlindan, James
McAlindan, John
McAlindan, John
McAlindan, John
McAlindan, Patrick
McAlindan, Patrick
McAlinden, Arthur
McAlinden, Bernard
McAlinden, Denis
McAlinden, Elizabeth
McAlinden, Henry
McAlinden, James
McAlinden, Michael
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A separate question to think about: Did any of the McLindens have boarders in their households on the censuses? What were their names, where were they from, etc? That might help you trace back your family, since the boarders, if any, might have come from the same part of Ireland.
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That is about all I can think of. Have you tried the County Down Genealogy Group at facebook?
https://www.facebook.com/groups/CountyDownGenealogy/If the search ends up in Kilkeel, try A Day in the Life of Kilkeel & Mourne at
https://www.facebook.com/groups/436608986386995/Or maybe there is a local Clonduff Parish group.