Apologies OP as someone may have already said this, and I've just quickly skim read, but I wonder if it's your ex that's keen on having his girlfriend there because he's not used to parenting alone?
I've got 2 step kids, and I think an awful lot of them, but DH and I used to have some shocking rows about them because we were coming at it from different angles.
He used to have them every other weekend (they're adults now) so whilst I'd be around to say hi and have a ten minute catch up, I didn't join in with days out, cinema trips, family visits etc because they were coming to see their dad and I didn't want to encroach on that. Years ago I'd been a child who never got to see her dad alone, so really felt like it mattered that I used that experience.
DH, however, wanted me there because he just wasn't used to thinking up ideas or being around them without another adult. He felt I was coming over as not wanting to be involved, which was utter rubbish. A child shouldn't have to say they want some time with just their parent, it should be a given. When my DSC said they'd like me to do things then I absolutely did and loved their company, but I never wanted them to feel like they had to share what limited time they had with their dad, when I got to see him every day.
Women who drip all over their new partners whilst some poor kiddies are still struggling to process that mum and dad aren't together any more, In my opinion need to grow up.
I hope it works out for you and your Ex starts listening, as you know it's about the DC not him, hopefully he'll realise this before the damage becomes irreparable.