It'd certainly make this process easier to make them start talking about everything. It's not entirely graceful however, given that there will no doubt be negative consequences to the one that isn't Kira, and since obviously it was outed to everyone, that death wouldn't be incriminating to anyone unlike Rajam. Free kill. I'd say that they could out their role instead, but it undoubtedly has connections to their identity so it's basically the same thing to anyone who knows or looks up the source material.
Of course, let's say we progress without doing this, successfully get our votes on the actual Kira, Day is ending, what does the accused do? Upon reflection I fear that the same result might happen regardless, albeit with less certainty, due to a Kira-with-nothing-to-lose will out whatever they think might help anyone with the same win condition. Therefore perhaps there is some merit to sacrificing one to help get the other. With two whole days of lynching behind us, a third spent killing a townie is scary, even with the "quiet" Nights.
We may want to wait on people rereading/returning? If we find anything definite there we might be able to forego this necessity.