For what it's worth (not saying it's worth much) on the show, when Jaime and Bronn were on the beach on the Dornish coast and are interrogated by the mounted Dornish warriors, one of the Dornishmen says:
'You are from King's Landing.'
To which Bronn replies:
'Accent gave me away? Fleabottom, whelped and whipped.'
Bronn says this in his normal voice - without trying to disguise his accent - and if the Dornish warrior who spoke can be trusted to know a King's Landing accent, then at least in the universe of the show there's a bit of evidence that he's from KL. Which wouldn't surprise me at all.