Is Lord Rama right to renounce Devi Sita publicly the second time?
I think that the most traditional read on this question would be to answer in the negative and suggest that Lord Rama is wrong for asking Devi Sita to publicly defend herself. The logic here would be that Lord Rama was wrong to cast aspersions on his wife. His paranoia and sense of doubt about her compelled him to continually ask her to defend herself. This becomes the traditional read on Lord Rama's actions.
I would like to posit a different line of thought. Lord Rama is the embodiment on how dharma, or one's duty, must always be followed. It is tough, but not impossible to do. It requires one to strive to the highest of ends, and often do what is not exactly preferred. Dharma oftentimes is meant to be uncomfortably. Lord Rama himself embodies this with his willingness to be banished in the forest. He knows the command was made in a wrong light, yet he adhered to his dharma as a son. In much the same way, he recognizes that the embodiment of good leadership is to hear the concerns of one's citizens. Lord Rama believed in the lack of self in terms of being a leader. He recognized that quality leadership is where leaders subjugate their own identities for their people. Leadership is an ascension to the divine realm, and Lord Rama recognizes the duty of every leader to listen to his people. In this, he does what is difficult and what is right in accordance to the duty of leadership. Lord Rama does not do what is best for himself. It is here where his denunciation was done. I do not think that, after demonstrating the highest of tendencies throughout his life, he would suddenly capitulate to the lowest possible level, something reflective of base emotions and the worst in human nature. Lord Rama believes in the concept of teaching lessons through his example and this might be one of those moments that shows how political leadership must embrace that which is difficult in the name of the people being represented.