Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. So you probably won’t find statutory birth, death or marriage certificates in Ireland for this family. For earlier years you usually need to rely on church records, where they exist. Ideally you need to know the precise denomination and have some idea of where the person was born in order to search the correct records. Not all churches have records for that period and not all are on-line.
RC records are mostly on-line on the nli site:
For other denominations, the churches usually hold the originals but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
Church of England, in Ireland, is the Church of Ireland (COI).
This is what records exist for Drumcree COI:
Baptisms, 1788-1906; marriages, 1802-1903; burials, 1804-1903; vestry minutes, 1767-1941, which include poor lists of Protestants and ‘Romans’ in receipt of aid from the parish, 1765-1827.
Some of the above records are on the rootsireland site (I don't know what years) but for the full set, you need to go to PRONI.
There’s no parish named Drummore in Co Armagh. A complete list of the 28 parishes in the county is attached:
Again all their surviving records can be found in PRONI. It also has the COI records for Co. Cavan.
The 1911 census had 4 people named Bertram in Armagh. A labourer in Portadown and a grocer in Tamnaghvelton (parish of Kilmore). Also a Bartrem couple in Richhill, weavers. All were COI.
In Cavan there were 2 households. A retired grocer in Corronagh born in Co. Meath and his wife (place of birth not recorded), and a farmer in Corgarry.
No Bartrams in either county. So the name is not common in the area today. The name is not listed in MacLysaght’s “The Surnames of Ireland” but the location and religious denomination point to it being a Plantation name. So the family probably arrived from England or Scotland in the 1600s.
PRONI has this holding under ref T2660/1 covering the years 1798-1821 which mentions the Bertram families and military service:
Bundle of 11 documents comprising letters from Capt. Andrew Ferguson of the Dragoon Guards to members of the Baird and Bertram families, topics including domestic matters; birthday celebrations for the Prince of Wales, 1806; army life; and military engagements, including reference to the situation in Co. Wexford, Col Walpole, and Lord Cornwallis whilst serving in Ireland during the 1798 Rebellion, and Gen. Crawford, Sir Sl. Auchmuly and Gen. Whitelock whilst serving in Buenos Aires and Monte Video during the early 1800s. Also includes correspondence relating to John Bertram and his medical career.
T2660/1/4 has this, dated 27.11.1801:
Andrew Ferguson to his aunt Mrs Bertram Linlithgow Scotland
Letter ".... I have found the Regiment very agreeable hither to & not one exceptionable character amongst the officers and as for the duty of the Regiment no man could conduct it with greater mildness than Col. Goldie and on all occasions that I have seen him use his authority he has shewn great indulgence to the little faults of both officers & men. Genl. Payne last week inspected the horses & selected the best for the peace Establishment which has kept us all busy for this sometime past he could give us no information when the reduction was to take place or the number of Troops that would be reduced. We rather think that 4 Troops will be reduced & 6 continued if that happens Col. Goldie will be put on half pay which we are all sorry for. I am out of all danger & have determined to continue let the consequences be what they may at present the Mess exceeds what I can afford but another War may break out or the peace may reduce the expence of living - We have rec'd an order permitting one third of each Rank of the officers to go upon leave of absence. I wish to remain with the Regt. all winter & by that time the peace Establishment will certainly be known which all of us are anxious about & also before quitting the Regt a second time I should like to be better acquainted with the different duties - ..."