Wouldn't it have something to do with the malting process? Back 200 years ago, they would have used peat to fire the furnaces and kiln the grain. All of the malt would have been slightly, but unintentionally, peat smoky, right?
By the early 19th century, improved indirectly heated kilns meant that malt didn't pick up (much) smoke character, regardless of the fuel used. Also, by that time, brewers were doing everything they could to avoid smoky character in their beer, since it was considered to be a fault. For example, in the early 18th century (~300 years ago), one of the reasons why porter was aged was to give time for the smoke character (from "blown" brown malt) to drop a bit.
That said, anything is possible. Locally-produced malt used for privately-brewed ("house brewed") beer could have had some peat character, but I can't imagine that the big Scottish breweries in Alloa, Edinburgh and elsewhere would have welcomed it.
More to the point, Scottish "shilling ales" and "wee heavy" are based on late 20th century examples. By that time, there was no way that Scottish brewers were using peat in their maltings, nor were they using water with a peat character in their brewing. It would be about as likely as one of the big Munich breweries "accidentally" using beechwood smoked rauchmalt in one of their lagers.