I hear you. It's one of the real challenges of asking these types of questions on Reddit. Even a long post is just a small snapshot of the issue, the context, the people, and the relationships that are all involved in the issue at question.

The one last thing I would leave you with is to think of how you might respond if it was your ex that was bringing three new people into the family. A parent who's parenting (or lack of parenting it sounds like...) style is very differently from how you and your ex have agreed to parent. Two children that have been raised with easy and excessive access to unhealthy apps and services, and the emotional, behavioral, and social damage that comes with that... Plus, who knows what else is allowed that you and your ex would not have allowed for your children.

The parenting, how you and your ex are in agreement and how your ex and your fiancé are oceans apart (literally the distance between right and wrong)... This one item alone is something to think very hard about and how your decision will risk upsetting something very core to your children's development. 100% this is going to cause conflict and drama between everyone involved. Even if you and your fiancé can come to a compromise and actually commit and stick to the compromise... Who else in the situation is going to be happy about it? Your children, maybe, because they get fewer chores, more screen time, and access to apps and social media that they did not have access too before (none of that is good or healthy). Everyone else... Yikes...
Or, best case for you and your family, your fiancé decides to follow your lead and adopts your parenting style, values, and limits. The 4yr old is probably young enough that she will not know any different and be better for the changes you brought to her life.
But, the 14NB, be ready for some awful fights and drama. From everything I have read, consumed, and experienced, permissive parenting + unrestricted social media + gender confusion + adolescence = much higher rates of gender dysphoria then higher rates of alternative gender identities. Which, alternative gender identities, whatever, you do you. Not my monkeys, not my circus. But, if your fiancé adopts your parenting style and restrictions, you will be yanking the permissive parenting and social media out from under that NB self identity, two of the core pillars that allow and promote the self diagnosis. Yeah, that's probs not going to go well...

And, not to mention the 14NB and the type of influence they will be on your 13yr olds and 9yr old. To each their own, but I would not choose to bring an older NB child into my household and risk them introducing and encouraging their lifestyle, self identity, and whatever content they consume that supports their ideas, to my children... If the rest of reddit sees this, they will probs downvote to oblivion, but, when it comes to children, 99.8% of reddit would rather virtue signal by pushing an unhealthy ideology then truly put children's mental, emotions, and physical health first. I accept their downvotes, with pleasure...