Installing bigger jets and/or smaller diameter metering rods will enrichen the carb fuel/air mixture. Sounds like that's what you need. The adjustment for the air door is just to tailor how fast the secondaries open. If you loosen it the stumble could be worse since it's already running lean. Get the jetting correct and then adjust the door so it opens as quick as possible without any stumble.
If you look in the tuning manual it will show which jets and metering rods were installed in the carb and will have a chart that you can follow to jet it up. Of course you'll need a jet/tuning kit to do this with. Jetting it up 2-3 notches and the stumble will probably go away. Remember to jet the secondaries up the same amount as you do the primaries to keep it balanced. In other words if you jet up the primaries 3 sizes jet up the secondaries 2 sizes also.