6 shillings is odd, I dont remember seeing this for another bank. I guess the denominations printed by the banks were what was needed at the time by commerce.
Lower value notes were intended to compensate for the chronic shortage of coinage which resulted from the Napoleonic Wars. According to PMI, 'silver notes' were defined in an Act of 1799 as notes for 9 shillings, 6 shillings and 3 shillings 9 and half pence. The last one equates to 1/6th of a British Guinea. The non-equivalence of the Irish and British pounds lead to odd denominations in Irish currency.
It seems likely that the bank of John Joyce & Co. failed in 1814 in the fall out surrounding the collapse of Ffrench’s Bank.
In 1824, Hely Dutton states that:
Of Mr. Joyce's (Galway) debts, I believe almost ten shillings in the pound have been paid, and it was the general opinion, that from his highly honourable character, activity, and skill in business, had he lived, the remainder would in a few years have been paid.