Help me catch him

So one of my closest friends is a guy. We've been friends for 2 years now, gone on lake trips and bike rides and ghost hunting in abandoned buildings and go country swing dancing every week. I'm kind of a tomboy so I do have lots of male friends, nbd. I always knew this friend (we'll just call him R) was kind of a dick, but we (my other friends and I) saw a positive change in his attitude when he left a toxic relationship.

So >> a few months and he meets this girl, we'll call her K. Since I'm his designated girl best friend, he values my opinion and asks me my view from a girls perspective. K and I end up becoming super good friends, she's super cool with me, they start dating, and everything is great for like a month. And then the fights start. Of course at first I took Rs side, because his side was the only one that I had known for 2 years. But the more I get to know K and the more I hear her side, the more I realize that R is manipulative, controlling, hot headed, and verbally abusive.

Pretty soon their fights make it out in public and I'm stuck in the middle. Long story short, now R has his lies confused, has lies on top of lies and has his timeline messed up. His story keeps changing. K thinks he's cheating, and tbh after all of this I wouldn't dismiss the possibility.

However, K, being one of the most generous and loving people I've met, doesn't want to jump to conclusions. I've been trying to tell her to leave his psycho ass for months, even told her I'm basically done being his friend because I'm seeing his true colors and they are ugly, but she wants definitive proof before she'll leave him.

So my question is: how do I catch a cheater? I've never been cheated on (at least to my knowledge) and I've never cheated.

Or better yet, how do I convince K to leave a toxic relationship with a controlling and emotionally/verbally abusive asshole who spreads lies about everyone he knows?

I myself am ready to cut ties with him the minute he tries to involve me more than I already am or disrespects me.