Is the flue is clear all the way up, including the roof cap?
Does the damper control both interior and exterior fireboxes?
My guess is too small a fire in the exterior grill area to get the chimney (flue) heated sufficiently to draw properly. Or another way to look at it, is that the main chimney flue was originally sized for a larger traditional, full multiple (3-??) split fire, for warmth and entertainment for the interior.
A small cooking fire may not be generating enough heat have the exhaust gasses not be cooled prior to exiting. If not enough heat to push upward, "heavy" air and wind can exert a resistance, dropping smoke, and presenting it into your interior space.
I would try a larger exterior fire, at least as an experiment. Or look into a smaller insulated flue, dropped inside the original chimney, that could retain enough heat to keep the gasses rising through the roof cap. Assuming that the interior fireplace will retain the gas logs, and not be returned to a wood burning status.
A mason should be able to isolate the two fireboxes, and install a properly sized drop-in insulated flue that would serve only the exterior.
May be as simple as a birds nest, especially if the interior fireplace was not used with wood for a number of years. And the exterior floor firebricks appear to not have had much use as well.
Cool set-up. Hope you can get it functioning the way you would like.
