My guess is your rock contains epidote that gives it the green color, but also other minerals as quartz, feldspars, amphiboles and/or carbonates.
You can find likely rounded rocks containing massive epidote at this webpage.
Epidote in unakite: Tumbled stones made from unakite, an igneous rock composed mainly of green epidote, pink orthoclase feldspar, and quartz. This unakite was mined in South Africa.
Epidote's density is a bit higgger than the value you provided: $3,4 g/cm^3$.
Your rock does not appear to have pink feldspar but it could contain quartz, feldspar or carbonates. As density of those minerals rounds $2,7 g/cm^3$, the resulting density for the rock would match with your measure.
The mindat description for Inishcrone beach says there is a dolerite dyke in contact with the carboniferous limestones of the area.
Epidote appears at dolerite dykes.
My guess is it comes from the dyke, or from a surrounding skarn system.
A skarn is an hydrothermal alteration of carbonates, that would also match with the geology of the zone, described in this old paper. This alteration appears when fluids of igneous origin (maybe associated with the dyke) increase the temperature of the limestones they cross.
There is also a gabbro intrusion at the west side of the bay. It is unlikely to find epidote on igneous bodies but the rounded rock could have been transported from the intrusion. My guess is it is too much faraway to be the origin and epidote is rare on gabbros as said.
Geology of Irishcrone site, Ireland. Bedrock Geology 100K. Geological Survey Ireland.
To determine the origin you would need to identify the minerals that go along with the epidote. A fresh-cut and a picture with a magnifying glass could clarify it.