Evidence has shown that random screening tests are effective in reducing COVID-19 infections in schools. However, test administration may be hindered due to a limited budget or low participation caused by pandemic fatigue. Thus, we seek to balance the number of tests administered with end-of-semester infections. To do this we use an SEIR model to simulate SARS-CoV-2 transmissions within K-12 schools, design a multi-objective simulation optimization problem, and tune an NSGA-II algorithm to find the best testing schedules. We find the Pareto front of optimal schedules of screening tests, which can be used by stakeholders to inform test administration strategies. We discuss insights about the characteristics of optimal strategies, for example, when there are limited number of tests available or a desire to use few tests, the optimal plan is to perform the tests earlier in the semester and at higher intensity.