I have said the best course of action for him is to be the best parent he can be and to put the kids first. I believe that this will give him the best relationship with his children and do right by them.
What's the alternative? Have a long running cold war with his ex/the partner on something he cannot control or police? Put the kids in the middle where by they will upset either mum or dad? Make them chose? How does that serve the children? How does that serve his relationship with them?
I don't think the ex has chosen to do the right thing here and the children should have been able to choose what they call her partner and the default should have been [first name].
But I don't see how the OP telling the kids they ought not to, or making them feel even more guilty than they might otherwise do for hurting his feelings or being disloyal to him will help?
Whether the OP likes it or not, this man is in their lives - perhaps fleetingly, perhaps permanently. He doesn't know and they don't know. But I do think his focus should be on being there as a dad, listening, validating and supporting the kids and to not try to let the OM or whatever his ex is doing/saying get in the way of that.
Different families manage the step relationship differently, I'm sorry if the idea that his children might have a close relationship with their mother's partner over the years and may call him dad as well as their dad was appalling to you. I didn't mean it as a replacement.
But ideally in the future, should his daughter marry, you'd like her to not be stressed about her mother, father and step dad all being in the same room together because she has experience of them all acting like grownups who put her first rather than fighting over her and leaving her feeling anxious and guilty for something she couldn't control.
I'm not saying that he should encourage the children labelling of the partner as dad either, just that it shouldn't be a major source of his attention when speaking to them. He could listen to their thoughts about it if they express them but I don't think you can really put it on the kids that it's their job to not do it and leave them caught in the middle with anxiety and guilt and worrying about the feelings of the grownups who are meant to be looking after them. 🤷♀️