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Thanks for posting
Does the engine use any coolant, do you have to add coolant or water to the reservoir tank?
Is there any kind of smoke from the exhaust when cold starting?
Have you tried replacing the spark plugs?
What you are describing about running rough when first starting sounds like a headgasket that is starting to leak internally, it doesn't take much coolant in the combustion chamber to cause a misfire, it disappears when the engine gets hot because of the metal expansion of the head and block. This can be checked by doing a block leak test using a chemical or with a diagnostic machine like a emission control probe to check for excessive hydrocarbon emissions while the rough running is present, sounds like the dealer is not too interested in helping you so I would contact a decent repair shop that can do this, I would not be surprised if it is a headgasket due to the supercharger which puts a lot more pressures and heat into the combustion chambers. At this time you can keep driving it and see if it gets worse or to be sure just try a new set of spark plugs, at the same time inspect the plugs for any that look different than the rest, on this engine cylinder #6 usualy is where the gasket blows first.
It seems that's the way it is anymore, unless there is a check engine light on they either don't want to be involved and or don't know what else to do, the self diagnostics are pretty sophisticated but not all problems turn a check engine light on and some common sense, experience and a desire to help a customer would be nice. Best of luck.