My name is ***** ***** I will do my best to help you with your BMW.
I have seen this many times. The sensors did need to be replaced but sometimes when the sensors fail it also damages the ABS control unit. The fix is going to be to replace the ABS control unit or have it repaired. There are some shops online that you can send out the ABS unit and they will do a repair on it for you to fix this problem.
This is an example of such a website.
I hope this information is helpful. I have replaced many ABS modules for this exact problem. Please remember to provide positive feedback as this is the only way I receive credit for helping.