According to crime historian and professional Irishman Owen Dudley Edwards, William Burke took the sacrament of holy absolution while in Calton Gaol. It was administered by a Father William Reid, who apparently served for many years as assistant to the Vicar-General for Scotland* before retiring to a parish in Dumfries, where he died in the 1840s.
I've done my best to find authentic likenesses of all the background characters in the Burke and Hare comic, but inevitably there's quite a few who remain maddeningly obscure, and in such cases I've had to improvise.
*The Catholic episcopacy in Scotland had been abolished during the Reformation, and would not be recreated until 1878. Ironically, this meant that the surviving Catholic mission in the country was more dependent on direct Papal authority, not less. Doh!