Good job on the picture.
Yes, it looks like a 4"vent but it only exposes 1/2 the hole opening. It is not unusual to have a little smoke seeping out where the BSG enters the cabinet, particularly when mounted to a foreign box. You should not have smoke exiting through the body of the BSG and out the puck tube. As I mentioned that will gum up the switch and puck pushing mechanism giving you grief.
First thing I would try is ensuring that wind is not playing a factor in preventing flow of smoke. It does not take much breeze across the top of the vent to cause back pressure. Also, make a 6" stove pipe out of anything you have, large tin can, rolled up cardboard and temporarily tape it on you vent outlet. Do not use cardboard permanently...FIRE HAZARD!! Just try it to see if it improves flow.
Failing that try to remove some material from the vent opening to enlarge it. It appears you only have control over half of the diameter of the opening. It would be easy to change that to adjustable fully open or fully closed.