Bmw aren’t bad for keys, I had to get one for my bike couple years back. Think it was around £50 and took a couple of days to arrive from Germany. This was a 2011 bike.
However, you need to take the bike to the dealer as they are required to take pictures of the bike and numbers. They also as you say require V5 and proof of address for you to match V5. All as to be expected for security, your issue is getting bike to them.
So probably easier if you have the bike on a trailer would be take bike and current owner to the nearest dealer. Get them to order the key and this will also be a good test to how legitimate the current seller is. If they are above board they won’t have any issue doing this. If not walk away would be my thoughts.
Car vertical is great for checking bikes and cars used them a last few times.