How does having students working at part time jobs interfere with their reputation?
If you can't hang out after school, when would you?
Besides, how is anything the students do after school outside the school's property part its business? It isn't a boarding school.
Well, from what I gathered in my time in japan, the issue is more that they want the kids to go straight back to their home so they change out of their uniform and back in the care of their parents before doing whatever they do.
The uniform is really the "issue" there. We can see that Hana is wearing it when changing out of her working uniform to go back home. And the uniform kind of advertise for the school. If parents see some kids wearing a school uniform while going to fast food or arcade after school, they might think that the school is too lenient and don't have "proper" students, then think twice about sending their own kids there. So yeah, it's more about the kids having "bad" behavior while wearing their school uniform. Once they're back to their place and wearing their own clothes, they have less chances to be related to the school.
It might seems stupid to us westerners, but education and school is quite a big deal in japan.
As for the ban on part time job, not quite sure, might be because seeing Hana walking home late, people might get strange idea as for the reason why she was out so late... And if comuting takes too much time, students can't go back to change in casual clothes then go to their job in time.
Anyway, glad to see more of Morigana's work. Would be nice indeed to get a cameo of Mari and Akko, maybe as customers of the shop for their daughter ? :3