If you find a good fanfiction about Harry's life after Hogwarts (with canon ships) could you please tell me?
I think that Barty Jr. was pretty clever. He just wanted to make sure that Harry would participate at the Triwizard tournament. So by putting Harry's name under another school name that didn't even exist, Harry was chosen because only his name was there as the fourth school's student. On the other hand, if he had put Harry's name as a Hogwarts student, he could not be sure if Harry would be selected by the goblet because many other competent Hogwarts students had put their names in the goblet.