As I mentioned above, it is very easy to open the booking on finnair.com and change the FF number over. I do not think that making an award booking for another person gives the option to enter a FF number.
I've tried doing this, but doesn't seem to work. I can open the booking and change details such as phone numbers, but can delete the BA #, nor choose a different FFP. Does it matter that my booking was Avios + cash?