Hmmm, 3 manuals and 4 shops it still isn't running right, I'm not sure what to tell you. I'm no expert by any means.
I just know that until all of that stuff is checked and verified "good", you are shooting in the dark. We're no experts and have zero time lately and since this is our 1st foray into the 350 twins, we went to Paulages on it. He got it sorted, explained it all to us on how/what order to do it and I rode it home today. It purred like a kitten. It ran 1/2 decent before but wasn't quite right. It would do what yours is doing. It can be a few things but you need to set things up in the correct order or you will be chasing your tail. Since you haven't said that you've actually adjusted the bike the way it's supposed to be, just did a "general tune", I would say, go back and follow the manual and the sequence the way it needs to be.
You've already made a change outside of spec by adding pods to it without verifying that everything was okay to begin with. Since these carbs are pretty finicky and don't tend to play very nice, I just don't know what to think about putting pods on. I've seen it done but usually with jet or whole carb changes. So since you feel that putting more or less oil on makes a difference, how do people figure out how much to put on? I've never had them so I have absolutely no idea how to figure out stuff like that. I've thought about doing it to one of our CL350's but I've just decided to stick with stock airboxes.
If you think around $50 is expensive for airboxes on these, everyone else is over $10/airbox more. Sorry, I was trying to save you some $$ to begin with.
Oh, and clean your carbs again. and again. and again. 4th time for us was the charm and 2 of those was in an ultrasonic cleaner. Stubborn suckers.