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I would not take the car to a dealership in fear that a tech would do something idiotic, like flash in a stock tune to get rid of the codes. I've heard plenty of horror stories of customers specifically telling the service manager to tell his tech to leave the tune alone and just read the codes, only for the tech to put in a stock tune. Hello catastrophic failure that the dealer will not cover. Find a reputable LSx shop and had the car flat bedded there if you must
Metal expands when its hot, so the fact that you are only having the issue when the engine is cold suggests you have seals about to go (or are gone). I would highly highly highly advise doing a leakdown test as a minimum to check out your cylinders. If you have two side by side acting up -> blown head gasket. If its just one, maybe a stuck or bent valve.
I remain skeptical, given your initial indications and this bit of information.
I would get this car to a professional tuner before I went very far at all.
You have at least one indication that this engine may not have been put together correctly, and may pay greatlly later.
Why? Magnuson used those green 42 lb. injectors with those kits, which is a Ford Lightning injector if I'm not mistaken.
And rough running or misfires on cold-start can be caused by a bad injector. I've seen it a bunch of times. Would only show up on cold-start on the ones I've seen and diagnosed first hand. The code he had stored for the cylinder #1 fuel injector backs up this theory even more.
This would also cause the smoking problem on startup. Case closed, I'd say.