As for insignia, during my research I contacted the regiment and the CO himself sent me some info. My friend also has an original BD Blouse, with Lorne Scot insignia (if you look at the colour plates in my book, there is a photo of it). The formation badge was added in by myself for that shot.
I presume that the Lorne Scots platoons wore regimental flashes and the formation badge of the formation they wer assigned to. Look at the Reader‘s Digest 2 volume set on Canadians at War, and you will see a nice photo of Lorne Scots wearing diamond shaped Corps badges with their regimental titles.
More soldiers served in Europe as Lorne Scots than as members of any other regiment, so their history tells us.
I don‘t know if other platoons beside the 3rd Div HQ platoon ever traded off. I would presume they kept their own badges - in the frontline it was common to remove badges altogether, and since the steel helmet was worn instead of the balmoral/TOS, they wouldn‘t have stood out at all.
Charlie Martin says the Lorne Scots platoon acquitted itself well; it would be interesting to see what the QOR war diary had to say about it.