It's a lot more complicated than that. Under the Good Friday Agreement BBC programmes have to be made available to Republic of Ireland viewers. RTE and TG4 have to be available in Northern Ireland (to the point where they even broadcast on Freeview) As for funding BBC gets royalties from the Irish Cable companies and Sky. On top of all that BBC Northern Ireland has gotten Irish Government funds in the past for productions and usually gets an amount of the Irish Licence fee via the BAI Sound and Vision fund.
They must also care somewhat about what people in the republic think as they seem to enter every All Ireland Broadcasting award available. They're currently the Best television News programme in Ireland where they were competing with RTE, UTV Ireland, TG4, and TV3.
That isn't true about the Good Friday agreement. The only reference to broadcasting is "explore urgently with the relevant British authorities, and in co-operation with the Irish broadcasting authorities, the scope for achieving more widespread availability of Teilifis na Gaeilige in Northern Ireland;".
This led to a low powered transmission of TG4 as it by then was to Greater Belfast on analogue.
The later Memorandum of Understanding between the two governments mentioned facilitating RTE availability in NI and BBC availability in ROI on DTT
, however, the Irish government states
"The Memorandum commits the two Governments to facilitating the widespread availability of RTÉ services in Northern Ireland on a free-to-air basis and BBC services in Ireland on a paid for basis. The provision of BBC throughout Ireland is a commercial decision for BBC as, unlike RTÉ, BBC does not have a mandate to provide its services throughout the Island of Ireland.
The Memorandum also ensures the continuing widespread availability of TG4 in Northern Ireland following the digital switchover."