Since Basics of Sword Fighting was an introductory course that students from all departments took, the students were not allowed inside the Sword Department's Training Hall just yet.
Thankfully, that Right-winged General's daughter was not in any of these core classes. The reason for that was because her family was a newly named family and didn't have enough backing and history to be included in their group. As for why the Lun siblings could be there despite merchants being lower on the social ladder than farmers, was because they were the only children coming from an influential family in the Mingya Country, who heavily relied on Tianan for military protection. So, it was understandable why no one of a higher status and influence was there.