| Don't let them fob you off. I would agree that you should take it to a BMW/MINI specialist. See what you can get in writing.
I would also start investigating getting an independant report from an expert.
Well before the warranty period is up you should send a letter of complaint by recorded delivery to the garage and also cc. it to your local trading standards office. Make sure that you say that you've only done 300 miles etc and that the service manager said that they couldn't go too fast because of the traffic.
Get a copy of the warranty. Read it through and check that you comply with the notification terms.
Make notes now of the dates, times and names of people you have spoken to and seen at the garage. | 
