So today I decided to break open my a/c unit and dive in. I disassembled everything, broke free the old cap, soldered on the new cap and put it all back together. In short, it worked.
That DIY was very helpful and informative, very easy to follow with great pics.
The dealership quoted me about $400 in parts and labor to fix it. My cost was $0.49 and about half an hour of my time. The DIY says it takes 90 minutes, but it took me about 20. If you have never soldered before it might take you a bit longer.
It was a very easy fix, and basically the United Bimmer forums saved me $400. I love this place.