Your name gets entered into the system based on the way you listed it on the I-485 - the Chicago Lockbox folks do not cross with how it is on your passport or any other document you’ve submitted. Your bio notice will show up with the names swapped (as listed on your I-485). When you go for your bio, (the swapped name will not stop them from admitting you), you however need to point out the swapped names to the technician to ensure your background check gets run with both the correct format and the swapped version. Your IL will also have the swapped name listed, (the bio technician cannot update the name), you’ll inform the IO of the swapped names when you go in for your interview the IO will make the corrections at that point to ensure your name is correctly listed on your GC when it gets issued.
Fingers crossed you don’t get approved without an interview, because if that were to happen, the GC will show up with the swapped name. You’ll have to file and pay for a replacement card in that case.