Lots of great choices to be made. I agree that if playing Portrush, it could be worth staying locally and playing Portstewart and Castlerock. Both courses are pretty near. The Bushmills Inn is a good choice for comfortable but characterful accommodation, and if you like whiskey, so much the better. By the way, The Valley (at Portrush) is a great course too. Ballyliffin is about 1hr 45mins drive from Portrush.
I would say that for an authentic Irish experience, the Carne, Enniscrone, Rosses Point (Co. Sligo) leg is more soaked in local character and charm. Portmarnock is a great course, although not all love it, The Island is great, but Royal Dublin gets mixed reviews and is not very high on the list of 'must plays'.
I think the itinerary depends a bit on your arrival and departure points from Ireland, and how much time you have. If your trip needs to take you to or from Dublin Airport, then Baltray (Co. Louth) is worth considering. It is up the east coast about an hour from Dublin Airport, and could be included en route from RCD. If your arrival/departure point is Shannon, I would probably drop the Dublin leg entirely and stick to Northern Ireland and Carne etc.