OP, your last post clarifies more the position you're in. The one before that appeared somewhat as if you'd moved out and left the kids in the family home, coupled with your thread title didn't paint the situation in a great light, which is why I came to the conclusion I did - apologies.
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Firstly, you need to re-frame your position of having moved out and left the kids in the marital home. In truth you've moved out, but someone has to be the one to leave, and if you have 50/50 residency of the DC, then there's no reason why your house isn't as much their home as the one they shared with you and their dad.
The reality is that children attach far less value to something which is the family home than we as adults do. The value they hold is to the people who give them the most security and stability, and while at the moment that house may hold some value for them, as it changes, and if your ex ends up moving another woman and possibly even children in, the dynamic of that house will change anyway.
In my case I was the one who moved out, and me and eXH had 50/50 as well. But in truth DS spends far more time here than he does at XH's. And eXH now has a new partner, a stepdaughter and a baby, that house no longer represents the family home that we shared - iyswim.
WRT your STBX introducing another woman into the equation, I wouldn't say anything if he's not up for discussion on the matter. In truth there's nothing you can do really, so saying something to him is only going to add to your own sense of frustration. Instead try to channel that energy into being there for the DC, and making your house their home. On many levels you are in a fortunate position wrt your DS, because he is eleven and therefore able to make his own decisions, so if he doesn't like the set-up with the new woman then he will be able to vote with his feet and stay at yours more. I don't recall how old your youngest is, but again, if you are there for them to listen non judgementally, then in time they will make their own decisions, and opinions.
The PR element is difficult, because on the one hand I can see why it is relevant given that DS lives with eXH 50/50, but on the other hand, given he is about to make significant changes to his own set-up, I wouldn't be looking to change anything your end, that way if DS does react negatively to a new woman and children in his life, you will in fact be able to take back the control and enable him to stay with you.