There actually was NO PROOF Suzuya killed that cat or did anything. They just assumed he did it all since he wasn't acting "normal" and had a suspicious background.
Just goes to show these two weren't on the side of justice either. They were caught up in their own delusions about how the world was too.
(super spoiler ahead)
The guy that told the twins about his "Special" circumstance is the one actually killing everything and thought it was convenient to foist it all on some weird kid. He's also the guy who tortured Yamori (Jason).