The fact is you now have a large family with lots of kids to consider and me (I'm supposed to be your wife one day remember?). There are more people than child 1, child 2 and your ex to consider. Somehow you have to balance all of our needs because right now me and the boys are getting neglected
All of the above should be repeated back to her.
She knows full well what a dick your ex will be about re-scheduling contact so should be aware that it's only for genuine emergencies etc.
This isn't an emergency situation.
She put her foot down over normal contact with the ex, thereby reducing the chances of an amicable relationship.
Now she wants to click her fingers and expects you to ask a favour from her? 
where is her son's dad in all this? Does he have any input in their lives?
My guess is she wants you to be a father figure to her kids but at the expense of your own.
I think she's been emotionally blackmailing you for a long time and you just haven't noticed -
You no longer get to spend regular, quality time with just you and your dc - which you have previously stated is something you consider to be important both for you and your dc.
Instead you & your kids have had to make do with 'scraps' of time on a thursday and a 'once in a blue moon' weekend.
Your dp has had you spending more time with her dc, playing dad to them. You even took them all on holiday - without your kids!
Do you think she would have been happy if it was her boys that were not included in a family holiday?
Arranging a birthday party miles away which 'needs' your input should have been discussed with you, you are meant to be a partnership afterall.
However, as far as she's concerned you have no right to be involved in things like this - you're just there to be the dogsbody to make it happen......hence the lack of courtesy and respect in discussing it privately with you first.
and so what if your kids miss out on 1-2-1 time with you? She doesn't care,she expects you to prioritise her kids and her.
i wonder whether not allowing you time alone with your kids is a passive aggressive way of pushing them out of No.1 spot in your life?
It's not your fault her kids have a deadbeat dad.
You're already doing your best to give them a positive male role model in their lives.
Personally, i think you should take back 1-2-1 time alone with just you and your kids - don't allow your needs to be neglected.
I'd also rethink remaining in a relationship/getting married to a woman who doesn't give two hoots about your kids.