You mention searching records but it’s not clear what records those were. Have you checked the Scottish Poor Law records for Mary? If she was orphaned and destitute and sent to relatives in Ireland then that was probably done under the Scottish Poor Law. Their records (where they still exist) are pretty good and often contain details of next of kin, together with the relevant parish in Ireland. The records for Greenock appear to be in the Mitchell Library in Glasgow. They will do a search for you, for a small fee. You might want to search those if you have not already done so. See:
https://www.glasgowfamilyhistory.org.uk/ExploreRecords/Pages/Poor-Law.aspx
If Mary originated in Inishtrahull or somewhere nearby on the mainland then she should have been baptised in Clonca parish. Have you searched the parish records? (They start in 1856). The RC parish records are on-line free on the National Library site:
https://www.nli.ie/en/family-history-introduction.aspx
You mention that Mary might have changed her name to Doherty. Inishtrahull was not heavily populated. (In 1901, 14 households with a total population of 65). One Doherty household:
http://www.census.nationalarchives.ie/pages/1901/Donegal/Ardmalin/Inish_trahull/1184418/
Chances are she moved to the mainland for work.
The variations in spelling of the Loughrey surname are absolutely normal. Spelling varied all the time in Ireland. People were often barely literate but even when they weren’t, exact spelling wasn’t something they bothered about. The “gh” sound in Ireland tends to be spelled “ch” in Scotland as in Loch Lomond, and so that additional factor compounds the spelling options.