I had the exact problem with my 2005 360, namely running very rough on first cold start of the day, then clearing and starting normally for the rest of the day. I wanted to thank David and Taz for their advice on getting the fuel injectors professionally cleaned & re-O-ringed because this solved the problem. Interestingly, the dealer wasn't initially interested in pursuing this angle, suggesting that "they all do that, Sir." But he readily agreed to do the work and was pleased that it solved the problem. In fact, the mechanic specifically called out how well the car ran, suggesting to me that many cars of this era could benefit from this simple procedure.
Thanks to FN for posting the issue and for the detailed replies that solved my problem.