The amp is about £20 on eBay, the single wire is the alarm received signal post amplification.

The only worry for me is that the BMW wires are tight tight tight. 1.35m is the shortest cable they could use when robots assembled the car. Having a look at an X5 or E39 from the same era may glean a ~2m cable making installation over the existing harness a little easier. Try bmminiparts for discount new old stock.