You have this the wrong way around. Settle the RQ first. Once you know what you want to find out then think about the most appropriate research design.
But your notes on the question suggest you want to find out about a causal process, and therefore we know already that a case study is not appropriate (in fact it would be the least suitable of the common designs).
My first concern would be the presumption of the question. Do we know that racial dissonance is a cause? If so what more can you find out? If not your research will fail because the question assumes it.
Note the difference to:
What is the role of racial dissonance in…..?
Because the latter allows the answer no role at all. You must allow yourself to be surprised by your findings.
Then – your question and mine are both causal. So you need a causal research design.