White smoke indicates that coolant entered the combustion chamber. It could have been a head gasket, or a crack in the head or block, without seeing it or knowing more details.

BMW engines tend not to be tolerant of overheating; cooling system failures and overheating are often accompanied by major repairs ($$$)

While the oil separator issue is a possibility, it wouldn't be my first area of investigation.

One option is to buy a used engine from a wrecker.

Sorry but I don't know any shops in your area to recommend or point you at. I wouldn't go to the dealer for this, I would look to an independent shop with experience with BMW vehicles.

Good luck

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2007 X3 3.0si, 6 MT, Premium, White

Retired:
2008 535i, 6 MT, M Sport, Premium, Space Grey
2003 X5 3.0 Steptronic, Premium, Titanium Silver
2002 325xi 5 MT, Steel Grey
2004 Z4 3.0 Premium, Sport, SMG, Maldives Blue