Wexford County is an Irish County--there was some fighting there between Irish and troops loyal to the Brits in the late 1700's. The Brits won.
The Brits used the word "Royal" quite a bit to refer to military regiments and units.
So, my guess would be the button is a British button for the formation of a Militia in County Wexford, Ireland, after 1796.
FYI, that's a bit of guesswork on my part.