You can look up Irish aristocracy on certain websites. Or just try googling the Earl of Enniskillen and usually family is mentioned. That way you can confirm the information you have.
Burke's I think is the name of one those websites.
Griffith Valuation took place in Fermanagh in 1862
at that time a Henry Arthur Cole in the Parish of Kinawley owned 249 acres 21 rood and 34 perches which he rented out
There are other Cole's, 3 are tenants of the Earl of Enniskillen, Parish of Enniskillen.
By 1862 there are no Bowler's and only 1 Bowden in Griffith Valuation.
No James Cole even in the directories showing in the Enniskillen Dragoon Guards but there was a large military
Military message board for Ireland, google MILGENIRE
Military Genealogy Ireland.
There are records for the Military at the National Archives at Kew London, they have given their records to a pay website, find the link on the archive website.
I tried the directories and there are Cole's from Enniskillen listed from 1846-1894. I didn't see any advantage to checking them out as there would be nothing to connect the two families from the directories.
You should be able to look up the Earl of Enniskillen
(not a lot of family information more about the Estates)
ps just googled Earl of Enniskillen
William Cole was the first Viscount of Enniskillen, Florence
Court, Fermanagh (quite a pile) his father had been the 1st. Baronet and his name was John, his brother was Michael.
There is a list of the Earls, none were named James.