You don’t say what denomination the family were. However the only Burgess family in Fermanagh in the 1901 census was Church of Ireland so I’ll assume it was that. Dairies is in Devenish parish. A copy of the parish records is held in PRONI, Belfast. It might be worth searching that for the marriage, if that’s where you believe Mary Jane came from. (Tradition was to marry in the bride’s church, if she came from a different parish, and so that’s not necessarily always the same parish as where children might subsequently be baptised).
I don’t think the records are on-line anywhere and so a personal visit is required to access them. This is what PRONI has:
Devenish (Clogher diocese) Baptisms, 1706, 1798, 1800-23 and 1828-73; marriages, 1800-24 and 1829-61; burials, 1801-22 and 1828-1939; vestry minutes, 1739-1829.