MacLysaght’s “The Surnames of Ireland” says of Fletcher, “See under Lister.” Under Lister he says: “This name in Ireland has four distinct origins: 1 an abbreviation of MacAlister; 2 in Ulster the Scottish Mac an Leastair alias Fletcher; 3 an occasional synonym of St Leger in Co Kilkenny ; 4 English Lester (dyer) of Leycester.” To that, I’d be inclined to add that, if English, Fletcher could well be an occupational name. In this case from the French “flechier”, ie arrow maker.
You haven’t said what denomination your ancestor was. Knowing that may give a clue as to his more likely origin, especially if you can point to a precise denomination. What were they in Canada?
Looking at the 1901 census of Ireland for Co Antrim (where your ancestors’ lived) there were over 300 Fletchers then. Almost all were of one Protestant denomination or another, so that points strongly to them being incomers at some time in the past. A high percentage were Presbyterian which indicates likely Scottish origins, but there were many Church of Ireland and Methodists (which is a breakaway from the Church of Ireland) and that tends to point to English or Welsh roots. A detailed DNA test might tell you a bit more if you match with any British origins.
You say you cannot find any baptism records in Belfast. Not all Church records are on-line, especially for some Protestant denominations. If you knew what precise denomination the family was it might be possible to suggest which sets of records to look at in PRONI (the public record office) in Belfast. But a personal visit is required to view them.
You mention his wife’s surname was Esten. That is not a common name in Ireland. In the 1901 census, in the whole country, there was just one. He lived in Belfast but was born in England. He was Church of England (which is the same as Church of Ireland).
But as an overall comment, it does seem fairly likely to me that your ancestors are descended from folk who probably moved to Ireland in the 1600s. Possibly as part of the Plantation (though Co Antrim wasn’t involved in the Plantation because so many folk had already moved there, particularly from Scotland, for other reasons) and there were a few waves of later movement due to things lie famine in Scotland in the 1690s. There are no Fletchers in Co Antrim in the 1630 Muster Rolls so that suggests an arrival after that period.