Tempting but, no, I wouldn't do it - I would do as faothinthesound says.
My MIL (and FIL!) are the same. I learnt (far too slowly) to not engage with it at all. They've, very successfully, managed to 'divide and conquer' DH and his sister by doing this. We're completely past the point of no return now I think. Anyway, I tried to gently suggest to SIL what may be going on over a decade ago now (when she shared how upset she was over the state of her relationship with her brother) by giving her an example of something FIL said (not badmouthing her - which he does do - but to show how they create rivalry). It didn't go well at all.
I can't think of one person that they haven't tried to badmouth to us and I have absolutely no doubt at all that they badmouth us. People work that out eventually.
They eventually worked out that they can't do it with me (they could to start with I am ashamed to admit - I just didn't get it) - I reflect it back, ask them if the person gave them permission to share their business, ask them how they know that, change the subject, say it is none of my business...but mainly just avoid them because it is utterly draining and I feel like everything I say will be judged, twisted and gossiped about.
Once I said to MIL in response to her saying something negative about her brother (again) - 'gosh I don't think I've ever heard you say anything positive about Bob - tell me some nice things about him' or wtte. It was met with cats bum face and then, thankfully, the DC burst in.