I don't think you need any particular "approach," and it is a perennial problem that books such as Creswell and Poth which emphasize such approaches fail to cover the most common way of doing qualitative research: semi-structured interviews followed by some form of thematic analysis.

In in this case, more important questions might be who you want to interview and whether you want to use individual or focus group interviews.

You can design literacy framework based on your study variables in form of preparing interview schedule (questions with scaling techniques - likert scale). Get into survey by interviewing the samples. keep some open ended questions for getting additional informations. If it goes as a case study, it will be more narrative and complex to embed with SDGs as you said.

I believe a case study is an appropriate method.