While the CS education community has successfully incorporated tech-ethics assignments and modules into computing courses, we lack a defined process for instructional design to create these materials from scratch across the curriculum. To enable the development of such a process, we explore two research questions: (1) What specific instructional design challenges emerge when creating ethically-integrated assignments for CS courses? And (2) what strategies might overcome them? We address these questions using Research through Design, a method for critically examining design processes. Applying this method to our own process of creating ethics-integrated CS assignments yielded four key challenges: identifying an ethical context, maintaining a technical focus, eliciting both ethical and technical thinking from students, and making the assignment practical for the classroom. Further, the Research through Design approach revealed process-level insights for addressing these challenges, which can apply across the computing curriculum. This paper also serves as a case study of Research through Design for CS education, highlighting the importance of the instructional design process and the behind-the-scenes challenges and design decisions that go into tech-ethics materials.