Hi Laura,
How exciting that your family are planning a roadtrip through Ireland.
Here you can find tourist-board approved campervan rentals, maybe you can inquire with any of these if it's possible to have a different drop-off point.
You said you already inquired with Bunk Campers, and on their
website it says you can pick your campervan up in Dublin and drop it in Belfast, and vice-versa.
From Dublin, it's very easy to reach Cork by public transport, if you dropped the camper in Dublin and wanted to explore Cork after that - there's a direct train between the two cities.
Let me know if there's anything else I can help you with - for example mapping public transport routes.
Le meas,
Alinde