That fan shroud is $60 online (I think it is
17111439108). It is held in place by 3 reusable plastic rivets. The fan does have to be unscrewed from the fan clutch to get access. Requires a 32mm wrench and some sort of holding tool to keep the clutch from spinning. Even if you are paying for labor, no fair mechanic would charge more than 1/4 hour to do this. A BMW shop should be able to do it in 5 minutes or less.
I don't know your tolerance for pain, but I wouldn't call a previous owner, have them call the shop, argue back and forth, arrange to drop off the vehicle, arrange to pickup it up, etc. to avoid paying for a $60 part.
In terms of performance, it would seem to barely matter overall. Some "gorilla glue" brand black duct tape could cover the hole if you are convinced you can't tolerate the 2% extra bypass around the fan...