If you don't want students to trade because some are better at it than others, you could still do an automated trading round at the end. Start with the students that got their lowest choice (or possibly even no listed choice at all), and check for all of their preferred schools if there are students there that would prefer to be at this student's school.

Every trade would improve efficiency, but because students have no control over it, it wouldn't hurt fairness.

Though listing popular schools high can also be prevented by banning down-trades. But that would still be unfair because skilled traders will still be better at it. Automating it would solve that problem.