Something like this *should* be possible as long as you are doing a bit for bit copy. At that point you are not doing any kind copying of files or directories but raw data. The best tool I know for for this sort of thing is dd which is found on almost any Linux OS. I think Linux is going to be an ideal tool for you here as generally speaking Linux has far more support for different types of files and file systems.
That said I have never tried a this my self so do so at your own risk. I would also say that dd can be a VERY dangerous tool if used incorrectly so read the documentation VERY carefully. If you are not careful you can write the contents of a new blank disk over the only copy of data.
Then you have to consider fit. While I can only assume that the drive BMW elected to use is of a standard size you could find your self surprised. I would pull the disk out of the car and learn as much as I could about it before spending my hard earned money on a disk that could be the wrong part.
The more interesting questions are how the storage works and how BMW feels about people doing this sort of thing (People tend to get butt hurt when you copy their closed source software). Based on the documentation posted earlier in this thread it looks like the system uses disk partitions to to know what data is where so one could possibly buy a SSD that is say 120 gigs and expand what would be normally be 8gigs reserved for media to 48 gigs.
All and all I would say that you are taking a gamble to the tune some where between 200 and 300 dollars depending on which drive you go with. One way another a lot of people here could benefit from your efforts so make sure and keep us updated with how it goes.