Example,
https://www.6post.com/forums/showthr...t=engine+sized
That person had his engine sized and it cost him 8-9k out of pocket at the end. His case is very much like you. He checked his oil and no low alerts. However, the BMW mechanic stated possible of lack of oil. Another case of a guy had his engine sized too and no alert of low oil either, but when they took out the oil, there was metal shaving in it. Again, lack of oil. Many more examples in the m5 and m6 threads if you do a search on "engine seized"
In your case, 2k out of pocket sounds like a good deal since the car is out of warranty. However, you have extended warranty cover, maybe ask them cover the 2k. GL