Don't know what year your '02 is, but the arm rest in question is, I believe, from an early RHD car. The metal plate, that has broken into pieces bolted to the arm rest and to the door, and, I think, the vinyl cover piece was a push-on type location on the metal plate. Can be seen on page 51/57 in the BMW parts book, or RealOem?.
Your driver side arm rest, with the screw, is the later type and as per the pics below and what I have on my Nov '73 build Touring. The BMW parts book does not (for some reason) show a picture of this type, but does give a different part no. from the earlier one.
If all else fails, you could post a want ad in the UK 2002 Forum.
http://www.02forum.co.uk/