The gunners trained in Canada at the BCATP schools mostly (nearly all) wound up in turrets on multi-engine aircraft. Like 6 Group. Single engine trainers were used because they were available. The Battle had been ordered by the RCAF as early as 1938, in small numbers. Large numbers of Battles were shipped to Canada to start up the BCATP, initially for fighter pilot training, but were found to be useless in this roll. (Couldn't perform aerobatics, for example). Converting Battles to Mk. ITT configuration was cheaper than building new multi-engine types, and used up airframes nobody else in Canada wanted. None of the turreted Battles were intended as gunnery trainers when they were built, they had all been new built as bombers or crew trainers. Several arrived in Canada with battle damage from the French campaign.
The Defiant was built in relatively small numbers, used precious RR engines, and continued in use in the UK, first as night fighters and then as target tugs as the fleet was used up. Introducing a new type in Canada would not have been beneficial.
The Lysander, and the Bolingbroke (which had largely replaced Battles by the end of the war) were, as you suggest, aircraft in production and available, and not much good for anything else. Both used Bristol Mercury engines which had a high failure rate in Canadian weather, but were available. Unlike Merlins. Engine availability was a problem for the RCAF at home throughout the war, and led to large government investments to build up capabilities at P&W Canada. Many of the Canadian built Hurricanes were shipped to the UK without engines, CC&F rotated a small number of available engines amongst the aircraft under test in Canada.
