An update - after 1 week the car does not start, but the lights seem to be working ok (for now). Decided to replace the alternator and saw the folowing on Ebay - which one would you choose? Thanks.

BMW X5 4.6is E53 £220 WATER-COOLED ALTERNATOR Bosch OE 150amp 2A167 ABBCO Luton | eBay

or

RTX Alternator For BMW 5 & 7 Series & X5 | eBay