Alexander Cochran

I have hit a brick wall for several years when trying to positively document the first Cochran ancestor of my family to come from County Antrim, Ireland. My issue is I cannot verify/find his family in Ireland for several reasons: I have no date of birth, hometown, obituary, immigration record, nor church records. 

Here is what I can verify about the Alexander Cochran I am looking for: 

Alexander was married on 29 June 1806 to Catharine Kirk (another brick wall!) from County Cavan, Ireland. They were married in Alexanderia, Hunderton County, New Jersey. Their marriage was recorded by the Justice of the Peace. He wrote that Alexander was from Antrim, Ireland. 

Alexander and Catharine had their first son, James in 1806 also. (I assumed James Cochran was in New Jersey, but I cannot find a birth certificate or baptismal record.)

Alexander became a naturalized citizen of the United States on 25 Oct 1808 in Hunterdon County, New Jersey. (This means that Alexander was a resident of the US beginning no later than 25 Oct 1803 to fulfill his 5 year residency with the one year immediately preceding his application residing in the state of NJ) 

Alexander’s Pennsylvania Veterans Burial Card, from serving in the War of 1812, does not have his birth year listed. Cannot find any record of any paperwork about his widow collecting his pension after he passed in 1822. (None of the military records in Fold3 or Ancestry have a birth date.) According to the Pennsylvania Archives, Series 2, Volume XII, Muster Rolls of the War of 1812 - 1814, page 547, Alexander was in the 32nd Regiment (Lt. Col. Commandant Conrad Krickbaum), 1stBrigade, 2nd Division of the Pennsylvania Militia under the order of the United States Army. Alexander was inducted as a Sergeant on August 21, 1812 and served under Captain Samuel Wilson as the 3rd Sergeant of five in the Regiment. 

Alexander and Catharine had their daughter, Maria Anna, baptized at St. John the Baptist Catholic Church in Haycock, Bucks, Pennsylvania on 26 Apr 1817. (From this it can be assumed that he was of the Catholic faith.)

The Federal Census of 1820 has Alexander listed between 26 and 45 years of age. Thus, his birth year would be between 1775 and 1794. (Being married in 1806, it can be assumed that Alexander would have been at least 16 at the time, it would further narrow the birth year to between 1775 and 1790.)

Alexander passed “suddenly” in December 1822. A newspaper article from 1900 discussing an investigation into the death of Alexander Cochran, a grandson of the aforementioned Alexander Cochran, disclosed that both his father and brother passed suddenly due to heart disease, insinuating that it runs in the family. 

Four Cochran children (John, Mathew, Alex and Sally) arrived on The Hannah 31 Oct 1801, they were accompanied by Alex Cochran, age 32 (birth year 1769). This Alex Cochran is not within the birth year range set from the Federal Census of 1820 and these 4 children are not with the Alex Cochran I am looking for any other time in the US.