Hi, welcome to Just Answer.
It's rare for the fan clutch to fail on these, unless the car has high milage...if the ac worked after the freon recharge, than the freon must have leaked out again. A quality shop will fill the system and check for leaks since freon is not supposed to leak out at all...you'll need to take it back to them for a recheck, but be prepared to possibly have a leaking a/c compressor or hose which are common failures on these. A compressor replacement on this car can runs upwards of $1000.
Let me get this straight...BEFORE you took the car in for service the a/c worked for 10 minutes - correct? Than you had it serviced and the a/c worked PERFECT for 2 days - correct? Now, it's back to working for only 10 minutes - correct?
If that is true than before you even had the car looked at and the problem initially began, was there ever periods of a couple of days where it worked perfectly?
Well it sure sounds like they realize they can't service your car correctly and are trying to ship you away to another repair facility. The freon does NOT need to be removed to change the fan clutch...the fan clutch on these rarely fail at 67k miles. If it was truly bad, they would not have any problem showing you the old one, and letting you take it to another facility for evaluation. Yes the car can overheat if the fan clutch failed, but you don't have any overheating problems and I hasn't replaced a fan clutch on a 3 series for overheating under 100k miles, it's still possible but very very rare. So honestly they should've only charged you an inspection fee of 1.0 hours and told you they can't conclude why the a/c isn't operational, and than recommended they send you to the dealer. But like typical "shady" shops, they charged you for an unnecessary repair and told you it was fixed. If you don't want to deal with them any longer, I would find what agencies in your state are responsible for policing the auto repair industry and file a formal complaint...or if they are a AAA certified shop, contact AAA and file a complaint. You may not get your money back any time soon, but at least they can't continue servicing vehicles incorrectly. If it ends up being just a programing issue, it would of cost you less then that at the dealer and you would have an operational a/c system.