Just looked it up. In 2003, the BMW ball mount was part number 72-11-0-009-120, labelled as E53 Ball. It was $79.50 CDN at the time, probably twice what a trailer store would have charged. I don't know if that part number has changed up or not.

On my original invoice, the hitch is the same part number as listed by the OP.

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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