Here's the information I received from the National Museum of Ireland-
"This looks like a very interesting item, my colleague Alex Ward has found that Byrne and Brenan, button manufacturers, were in business from 1795 to 1807 in Dublin. I cannot find mention of a Irish Brigade in the War of Independence. However this button might belong to a solider of the Irish Brigade (1794-98) which was created by the British Army and consisted of Irish Catholic soldiers from the French Army Irish Brigade (1698-1791), who had fled the French Revolution. They still supported the French King and served in 6 regiments of the Irish Brigade until 1798. But the way this was described as the Irish Brigade and not Royal Irish Brigade. The Irish protestant community opposed these regiments and they all served abroad mainly in the West Indies. For example the 3rd Regiment served in Jamaica and San Domingo."
So I was able to pin down the manufacture date between 1795 and 1807, but the mystery of who was the "Royal" Irish Brigade remains.