I think Berlin may have the winning card, if the liner is in a large clay chimney and there is clear unobstructed air passageway around the outside of the SS liner, then you could have a huge amount of draft pulling room air around the stove and up the chimney.
You can pack fiberglass insulation [insulation only no paper] around the fireplace damper where the liner goes through to block the air. Is there a sealed cap at the top of the liner attaching it to the masonry chimney??
Did you try the cigarette or candle trick yet?? I'm betting on bad gaskets around the front doors and an open fireplace damper around the liner.
The flame on the candle will be pulled sideways near these drafts or the cigarette smoke will disappear through the gaps.
Let us know what you find. Greg L