So, I think it's important that we (as the healthy half of the dynamic) try and remain grounded in these situations. We can either inflame the situation in turn causing more conflict, or we can reduce the conflict by remaining calm and centered. How much of this is actually about you being upset? How much of it is about you not liking the fact that you have this guy coming in trying to tell you how to raise your kids?
It's not all that uncommon that when a new spouse comes into a family, that there is this kind of ruffling of feathers about parenting. My husband doesn't like the way I parent my kids either, he likes to make big suggestions too.

Your ex very well may be trying to keep the peace in her relationship (common) and also is trying to have her boyfriend involved (common) so he feels included. She gets to do that, we can't tell other people how to live their lives.
We do get to have our own values in this and it's OK to set a boundaries in that. I personally don't discuss parenting matters with my husband's new spouse, I only talk about it with my exH. However, I do respect that they are a family and that
they will discuss matters that affect their family so if he needs to get back to me after he speaks with her - I get that.
Is there a chance that you can maybe step outside of these hard emotions? Perhaps from a calmer state, shelve the thinking that says "They are MY kids! And none of HIS business!" because it's not going to help your situation have less conflict. (Not judging here, I think I actually said these exact words myself) You can't control who she invites into those kids life, you can only control your reactions to him. In the end, you're their Dad. He can't change that fact and demanding that he not be a part of their life will probably prove to be a fruitless endeavor... . and it isn't really teaching your children a lesson that will help them. What happens when you re-partner? What precedent is being set in how new spouses are treated?
If it were me, I might set boundaries in this - that don't include emotional reactions to the new partner.
Ex,
I apologize for my overreacting recently. It's proven to be difficult for me accepting a third person's presence in our children's lives. So far it's only been you and me making these decisions together and has taken me a little off guard with his contributing to our discussions. I also understand that as their mom, you want what's best for them and see him as a positive influence. I think that's something we both have in common, wanting them to be healthy and happy, and I'm sure [your new partner] does as well.
In the future, I think that while it makes sense that you converse with [your new partner] about the decisions regarding the kids, the direct communication remain between you and me - being that we are the ones who are legally responsible for them and who are required to make the decisions in their best interest. We've done well with this so far and I'd like to continue with what has worked for us.
Respectfully,
Noise -DG