There are 2 songs in DT that mention the King's shilling:
MARCHING THROUGH ROCHESTER by Pete Coe – and there's another copy called COME BE A SOLDIER FOR MARLBORO AND ME
THE KING'S SHILLING from Jean Redpath

And I found 7 songs posted in the forum that mention the King's shilling:
THE SCARLET AND THE BLUE (Irish version)
THE SCARLET AND THE BLUE (English version)
I WILL GO from the Corries.
THE MAID OF MELROSE TOWN by Les Barker – and there's another copy here.
I'LL MAKE A MAN OF YOU
FAREWELL TO DERRY (where it is called a bob)

Unfortunately, none of them match the lyrics you have quoted.