I searched the subscription site Roots Ireland and was not able to locate a marriage record for a Bourke to a Morton. The Aglish (Castlebar) RC parish records start in 1824 for marriages and 1838 for baptisms. Here is a link to the parish register up to 1880 https://registers.nli.ie/parishes/1039 The Roots Ireland staff would have transcribed the Castlebar data from this register.
It is possible that Edward was from a parish near Castlebar that does not have records back to the early 1840s which would explain why a record is not available. Many emigrants would use the nearest big town when someone would ask where they were from
I checked the 1833 Tithe records for Mayo and the only Mortons listed were from Crossmolina civil parish to the north of Castlebar. Possibly Anna was from that area.
Any other information on your family? Have you taken an autosomal DNA test?
Let me know what questions you have.