Mother in law openly cheating

My wife (38) and my two kids (5 and 9) just spent a week with my MIL (62) at her holiday apartment together

I had some work todo and also didn’t want to be in a small 2 bedroom apartment with my MIL for a week so stayed at home

My first language is English and my wife speaks English and her own language as well as my kids while her Mum speaks only her native French language

About three weeks ago my MIL introduced us all at a family BBQ to her boyfriend (Paul) who she has been seeing for about 6-7 months, seems like a nice guy, she has been divorced from my wife’s step dad for about 16 years and there’s a few of these guys who appear, we get introduced to and then it fizzles out etc

I’m regularly speaking to my kids on the phone each night getting updates on what they have been up to in the day etc etc

My 9 year old tells me the story of how she met grandmothers boyfriend tonight and they were kissing

I wasn’t to happy about it as it’s only a small two bedroom apartment and my wife is on the couch so to have a man in there we’ve only just met once at a BBQ I wasnt best pleased

I ask if Paul (MIL boyfriends name) is staying there with them and my daughter says no it’s not him it’s her other boyfriend, he came to the restaurant with them and they were kissing

When my daughter has questioned it with my MIL (she’s 9 she asks a lot of questions) my MIL has told her that her boyfriend Paul hasn’t been very nice and is spending time with his grown up children and his ex wife and she has been upset and this other boyfriend makes her happy and it is fine and not a problem

I spoke to my wife next on the call and brought up what my daughter had said and my wife filled me in that Paul and MIL had been having problems and her FWB / boyfriend / lover (i don’t know what to call this guy) was messaging her Mum and she agreed to meet and he came for a drink at the end of the meal etc etc

I’ve said I can’t believe your Mum would bring someone like this around the family especially after just introducing her boyfriend

I got the response from my wife that she can’t control what her Mum does or what comes out of her Mums mouth before she had a chance to explain to my daughter.

I’m really annoyed by the whole thing and my wife just explains it like it’s not a problem and it’s over now anyway while I want to speak to my MIL and let her know that while she has the right to cheat and her boyfriend if she chooses to, do not do it around my kids and the. Don’t justify your shitty behaviour and make it seem like it’s a normal thing.

I already don’t have the best relationship with my MIL as our morals aren’t aligned and I’m quite reserved while she needs to be the centre of attention, she has a grip of my wife who won’t stand against her for anything.

But this is the second time that I feel the need to say something, first time a while ago my daughter asked why her and her husband broke up and I had to step in as the response was that he was awful, abusive, controlling and ran off with a younger woman - which saying any of that to a young kid blew my mind and I stopped her and let her know it wasn’t appropriate.

I’m just not sure if I confront her on this behavior as I feel I could blow up and tell her to keep her shady shit in the shadows away from my kids (who didn’t even know cheating on your partner was a thing that could happen) or just ignore it as another shitty thing she has done as it won’t happen again for a long time as there’s no holidays or kids spending time with her planned.

TLDR: Wife and kids are on vacation with MIL while I stayed home, MIL introduces her affair partner to wife and kids and justifies why she sees this guy to a puzzled 9 year old which is leaving me very annoyed