She might not be your kid but when she's under your roof you are responsible for her.
My brother is in a similar situation to your DP, and when the kids are with him and his partner they're expected to listen to them. I certainly wouldn't tolerate naughty behaviour towards his partner.
I do know his ex-wife was very bitter towards the whole thing and refuses to meet his partner/fiancee. Which is sad as between the 3 of them they're raising his two girls and hers.
I remember taking a cake to school once when I was in primary school mum made a slice and decorated it with hundred and thousands, and I had 1 for my party, but then I only had one party.
This kid is getting a whole lot of attention for a 5 year old. Money wont buy her affection, spending time with her will.