I think he is a Mole embedded in the Order by the Illuminati. But I don't know when, how or why.
It's not like Renzou has ever had strong convictions about sides in a conflict. He is the least loyal of the cram students. He has always been wishy-washy on his views. Heck, that is why he accepted Rin faster than Koneko. The only time he expressed strong anger was in his argument with Koneko. Koneko, like the rest of Myoudha, believe that everyone must do whatever they can to save Myoudha, even if it means sacrificing your own life. Renzou can't stand this. I think it's because of residual hatred towards Takezou, specifically what Takezou represented. In his opinion, Takezou must've been a fool because he died trying to save someone else. In Renzou's words, "if we die then there's no going back".
So why does Renzou work for the Illuminati? Does he believe in their ideology? Is he being coerced? Or maybe he just let himself get roped in? The central reason could be anything from money to for the lulz. It's frustrating, because nobody saw this coming.
Regarding possession: I don't think Renzou's strength is unprecedented. Renzou has always been agile and strong, useful traits for a Knight. As for his anti-demon power, I think Renzou was never forced to use that power before. Previously he relied on others to do most of the work. In the situation with Izumo, Renzou had to take quick action to disarm Takara and secure Izumo. Thus I think Renzou is not possessed, nor was he necessarily trained with additional skills.