1) The date on the source material is wrong. Possible but it draws from a chronological list of contemporary constructions, and in the absence of these papers and veracity of other parts of the journal I am drawn to trust the dating. If it is in number 19, then the 'Billiard Room' would seem the most obvious location.
2) The reference to the latter Bishop's Palace is a mistake. Possible given the assumption that Prioleau would have been involved. Was it even in the square, or at Prioleau's other residence in Allerton Hall?
3) The 'clubhouse' was in either 18a or 20 Abercromby square. Need to investigate the occupants (Harrison and McVicar), and also where a 'clubhouse' might realistically have been located in the houses.
4) A house, or maybe just a clubhouse, temporarily existed where no 19 is now for a year or so. It isn't impossible, and there has been anecdotal reference to a structure there being knocked down from Tom Sebrell ( American academic) but there is no record of it, and the plans drawn up by Culshaw's, including the levelling of the ground and the leashold layout, suggest not. It would however, make more sense than a 'clubhouse' being within someone's home perhaps. Intriguing possibility.
5) Something I haven't thought of yet.
Nest stage is going to be to try and locate membership and minute records ( if they exist) of the 'Southern Club' in this period. And try and locate more business records of the architect.