Stack effect has to do with the house pressures in relation to the chimney exhaust, some homes have a lower pressure that develops due to a local temp inversion or from appliances / air sealing which then want to steal air from another source, when its static usually the first make up air source is a chimney with a reverse draft, when a stove or insert is being used typically cold starts create a smoke issue, then the temps in the flue warm up enough and the flow of draft gets going in the correct direction, this can be mitigated with a make up air source into the main living space.
When I was saying leaky return, I'm suggesting that perhaps some sheet metal has not been correctly screwed to the furnace on the cold air return side, or a hole cut into the return in the furnace room for something (maybe the upstairs return at one point had a whistle or was to loud, so another hvac company hacked the return to fix the service call problem, but not the real issue)
Now to the BK stove specifically - needs to have an insulated liner in the chimney, this is due to much lower flue temps due to high efficiency, so if a normal insert was letting 750 deg flue gas exhaust on average, the BK is more like 400deg, you can literally put your hand over it on some settings, the insulated liner keeps those gases from cooling further with provides for a more stable draft, it wont fix your cold starts, but it will help with reloads.
Check the cap for debris blockage, grab some bino's and look, even a little bit of debris can cause and issue, before reloading the stove, open the t-stat fully & by-pass for 5 min, then crack open the door, see how if that helps with cutting the smoke roll out.