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The only James Allingham baptismal record on the subscription site Roots Ireland around 1805 is the Donegal record from 1802 shown below.
This link https://www.ireland.anglican.org/cmsfiles/pdf/AboutUs/library/registers/ParishRegisters/PARISHREGISTERS.pdf provides the status of Church of Ireland parish registers. Killasnett is shown as having records destroyed in the 1922 fire but some records were later found and may be in the Representative Church Library in Dublin. You may want to contact the library. I believe they will eventually digitize their holdings.
Here is a link to the Tithe listings in Co. Leitrim with the surname Allingham. A good number of records in Killasnet parish. http://titheapplotmentbooks.nationalarchives.ie/search/tab/results.jsp?surname=allingham&firstname=&county=Leitrim&townland=&parish=&search=Search&sort=&pageSize=&pager.offset=0
Name:James AllinghamDate of Birth:
Date of Baptism:27-Dec-1802Address:LagheyParish/District:DRUMHOMEGender:MaleCountyCo. Donegal
Denomination:Church Of Ireland
Father:Wiliam AllinghamMother:Jane LoveOccupation: