I think you could run a cold air intake channel with a damper in the bench so it opened into the feed tube above the flame front. This could connect to an external air source. As the system warmed it would warm the incoming air through the bench. Then you have a controlled draft in your house which has to pass through the thermal mass hopefully eliminating the idea that it is a "cold" draft. The pressure difference will always work itself out, at least this way you have some control over it. At least in my head
