Yes you can. I should know because when I first played KH2 it was in Beginner Mode and I had to unlock everything and I managed to get the secret ending. And when I played Standard Mode, I didn't need to complete 100% at all. I just needed to get a few things like beat Olympus Coliseum, I didn't need it to be 100%. And again, with Proud Mode you get the ending automatically.
Actually, you can't get it on beginner, I should know, I did 100% on Beginner; you probably just choose Standard on accident; and for Proud it isn't automatic, you have to get all the keyholes in each world, including 100 Acre Wood and Atlantica; Those are the only two ways to get it.