Actually,
you should check re the Visa holder. If they entered the UK through the usual channels then they should be OK for RoI and UK CTA. The reasoning being that travel to/from ROI via NI ( part of UK) is not controlled. So if you flew FR/EZY/BA to Belfast ( City or International) you could take a train/Bus/car to RoI without challenge.
It is worth checking with Immigration. I would ask, if the Visa is valid for UK would that satisfy CTA requirements?
I came across this which may also be useful.
Department of Foreign Affairs - Who needs a Visa?
It says that US citizens do not need a visa but do require a passport. (technically so do EU passport holders from UK as we are non Shengen)