The term "synthetic" when used to describe an oil filter or and air filter has no meaning whatsoever!

You have regular or 'dino' oil, and synthetic oil.

Use an OE filter, easy. Just go to a BMW dealer, or one of their online websites and buy the filter with a BMW part and BMW logo.

Change oil at TWICE the frequency the dashboard tells you, use a new filter, and you will be leaps and bounds ahead of any theoretic and unproven benefit that some 'special' filter will give you.